Properties

Label 32-273e16-1.1-c1e16-0-4
Degree $32$
Conductor $9.519\times 10^{38}$
Sign $1$
Analytic cond. $260037.$
Root an. cond. $1.47645$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·3-s − 10·4-s + 7-s + 4·8-s + 28·9-s − 2·11-s − 80·12-s + 5·13-s + 47·16-s + 4·17-s − 11·19-s + 8·21-s − 8·23-s + 32·24-s + 21·25-s + 48·27-s − 10·28-s + 15·29-s + 3·31-s − 46·32-s − 16·33-s − 280·36-s − 8·37-s + 40·39-s + 19·41-s + 11·43-s + 20·44-s + ⋯
L(s)  = 1  + 4.61·3-s − 5·4-s + 0.377·7-s + 1.41·8-s + 28/3·9-s − 0.603·11-s − 23.0·12-s + 1.38·13-s + 47/4·16-s + 0.970·17-s − 2.52·19-s + 1.74·21-s − 1.66·23-s + 6.53·24-s + 21/5·25-s + 9.23·27-s − 1.88·28-s + 2.78·29-s + 0.538·31-s − 8.13·32-s − 2.78·33-s − 46.6·36-s − 1.31·37-s + 6.40·39-s + 2.96·41-s + 1.67·43-s + 3.01·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(3^{16} \cdot 7^{16} \cdot 13^{16}\)
Sign: $1$
Analytic conductor: \(260037.\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{273} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 3^{16} \cdot 7^{16} \cdot 13^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(12.60500406\)
\(L(\frac12)\) \(\approx\) \(12.60500406\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - T + T^{2} )^{8} \)
7 \( 1 - T - 3 T^{2} + 2 T^{3} + 34 T^{4} - 122 T^{6} - 565 T^{7} + 3111 T^{8} - 565 p T^{9} - 122 p^{2} T^{10} + 34 p^{4} T^{12} + 2 p^{5} T^{13} - 3 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \)
13 \( 1 - 5 T + 6 T^{2} + 10 T^{3} + 19 p T^{4} - 720 T^{5} + 145 T^{6} - 4565 T^{7} + 64050 T^{8} - 4565 p T^{9} + 145 p^{2} T^{10} - 720 p^{3} T^{11} + 19 p^{5} T^{12} + 10 p^{5} T^{13} + 6 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \)
good2 \( ( 1 + 5 T^{2} - p T^{3} + 7 p T^{4} - 7 T^{5} + 17 p T^{6} - 15 T^{7} + 73 T^{8} - 15 p T^{9} + 17 p^{3} T^{10} - 7 p^{3} T^{11} + 7 p^{5} T^{12} - p^{6} T^{13} + 5 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
5 \( 1 - 21 T^{2} + 2 p T^{3} + 197 T^{4} - 163 T^{5} - 239 p T^{6} + 743 T^{7} + 6997 T^{8} + 33 p T^{9} - 10259 p T^{10} + 4408 T^{11} + 340012 T^{12} - 149669 T^{13} - 327916 p T^{14} + 505874 T^{15} + 7287304 T^{16} + 505874 p T^{17} - 327916 p^{3} T^{18} - 149669 p^{3} T^{19} + 340012 p^{4} T^{20} + 4408 p^{5} T^{21} - 10259 p^{7} T^{22} + 33 p^{8} T^{23} + 6997 p^{8} T^{24} + 743 p^{9} T^{25} - 239 p^{11} T^{26} - 163 p^{11} T^{27} + 197 p^{12} T^{28} + 2 p^{14} T^{29} - 21 p^{14} T^{30} + p^{16} T^{32} \)
11 \( 1 + 2 T - 43 T^{2} - 128 T^{3} + 751 T^{4} + 2911 T^{5} - 7079 T^{6} - 25011 T^{7} + 89586 T^{8} - 16491 T^{9} - 2184437 T^{10} + 146331 T^{11} + 34120480 T^{12} + 33194813 T^{13} - 280856670 T^{14} - 275502867 T^{15} + 1982899103 T^{16} - 275502867 p T^{17} - 280856670 p^{2} T^{18} + 33194813 p^{3} T^{19} + 34120480 p^{4} T^{20} + 146331 p^{5} T^{21} - 2184437 p^{6} T^{22} - 16491 p^{7} T^{23} + 89586 p^{8} T^{24} - 25011 p^{9} T^{25} - 7079 p^{10} T^{26} + 2911 p^{11} T^{27} + 751 p^{12} T^{28} - 128 p^{13} T^{29} - 43 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \)
17 \( ( 1 - 2 T + 4 p T^{2} - 146 T^{3} + 147 p T^{4} - 5560 T^{5} + 63345 T^{6} - 135302 T^{7} + 1216691 T^{8} - 135302 p T^{9} + 63345 p^{2} T^{10} - 5560 p^{3} T^{11} + 147 p^{5} T^{12} - 146 p^{5} T^{13} + 4 p^{7} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
19 \( 1 + 11 T - p T^{2} - 422 T^{3} + 1327 T^{4} + 13231 T^{5} - 52770 T^{6} - 286830 T^{7} + 1316435 T^{8} + 3604514 T^{9} - 28081056 T^{10} - 2271598 p T^{11} + 436051189 T^{12} + 420246053 T^{13} - 6706056535 T^{14} + 224221973 T^{15} + 142211799535 T^{16} + 224221973 p T^{17} - 6706056535 p^{2} T^{18} + 420246053 p^{3} T^{19} + 436051189 p^{4} T^{20} - 2271598 p^{6} T^{21} - 28081056 p^{6} T^{22} + 3604514 p^{7} T^{23} + 1316435 p^{8} T^{24} - 286830 p^{9} T^{25} - 52770 p^{10} T^{26} + 13231 p^{11} T^{27} + 1327 p^{12} T^{28} - 422 p^{13} T^{29} - p^{15} T^{30} + 11 p^{15} T^{31} + p^{16} T^{32} \)
23 \( ( 1 + 4 T + 134 T^{2} + 486 T^{3} + 8278 T^{4} + 27515 T^{5} + 13874 p T^{6} + 953642 T^{7} + 8621489 T^{8} + 953642 p T^{9} + 13874 p^{3} T^{10} + 27515 p^{3} T^{11} + 8278 p^{4} T^{12} + 486 p^{5} T^{13} + 134 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
29 \( 1 - 15 T - 6 T^{2} + 815 T^{3} + 2345 T^{4} - 56626 T^{5} - 990 p T^{6} + 61073 p T^{7} + 2048957 T^{8} - 60143926 T^{9} + 115849537 T^{10} + 850143313 T^{11} - 6898778608 T^{12} - 21046980820 T^{13} + 452641755556 T^{14} - 138544395442 T^{15} - 12857101724960 T^{16} - 138544395442 p T^{17} + 452641755556 p^{2} T^{18} - 21046980820 p^{3} T^{19} - 6898778608 p^{4} T^{20} + 850143313 p^{5} T^{21} + 115849537 p^{6} T^{22} - 60143926 p^{7} T^{23} + 2048957 p^{8} T^{24} + 61073 p^{10} T^{25} - 990 p^{11} T^{26} - 56626 p^{11} T^{27} + 2345 p^{12} T^{28} + 815 p^{13} T^{29} - 6 p^{14} T^{30} - 15 p^{15} T^{31} + p^{16} T^{32} \)
31 \( 1 - 3 T - 80 T^{2} + 753 T^{3} + 1092 T^{4} - 46457 T^{5} + 220332 T^{6} + 1046342 T^{7} - 14211319 T^{8} + 31888447 T^{9} + 369460130 T^{10} - 2912434243 T^{11} + 1186850305 T^{12} + 95815568717 T^{13} - 439310068799 T^{14} - 1217655157396 T^{15} + 18735868674340 T^{16} - 1217655157396 p T^{17} - 439310068799 p^{2} T^{18} + 95815568717 p^{3} T^{19} + 1186850305 p^{4} T^{20} - 2912434243 p^{5} T^{21} + 369460130 p^{6} T^{22} + 31888447 p^{7} T^{23} - 14211319 p^{8} T^{24} + 1046342 p^{9} T^{25} + 220332 p^{10} T^{26} - 46457 p^{11} T^{27} + 1092 p^{12} T^{28} + 753 p^{13} T^{29} - 80 p^{14} T^{30} - 3 p^{15} T^{31} + p^{16} T^{32} \)
37 \( ( 1 + 4 T + 177 T^{2} + 19 p T^{3} + 15864 T^{4} + 58980 T^{5} + 951780 T^{6} + 3131831 T^{7} + 41176929 T^{8} + 3131831 p T^{9} + 951780 p^{2} T^{10} + 58980 p^{3} T^{11} + 15864 p^{4} T^{12} + 19 p^{6} T^{13} + 177 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
41 \( 1 - 19 T + 55 T^{2} + 898 T^{3} - 5698 T^{4} + 16130 T^{5} - 337390 T^{6} + 2358667 T^{7} + 9439631 T^{8} - 161858061 T^{9} + 581239977 T^{10} - 3187530408 T^{11} + 37046289522 T^{12} - 51167015014 T^{13} - 1547466225765 T^{14} + 9408233383915 T^{15} - 34873463295586 T^{16} + 9408233383915 p T^{17} - 1547466225765 p^{2} T^{18} - 51167015014 p^{3} T^{19} + 37046289522 p^{4} T^{20} - 3187530408 p^{5} T^{21} + 581239977 p^{6} T^{22} - 161858061 p^{7} T^{23} + 9439631 p^{8} T^{24} + 2358667 p^{9} T^{25} - 337390 p^{10} T^{26} + 16130 p^{11} T^{27} - 5698 p^{12} T^{28} + 898 p^{13} T^{29} + 55 p^{14} T^{30} - 19 p^{15} T^{31} + p^{16} T^{32} \)
43 \( 1 - 11 T - 135 T^{2} + 2100 T^{3} + 6137 T^{4} - 176179 T^{5} - 40953 T^{6} + 8402878 T^{7} + 3448032 T^{8} - 271175414 T^{9} - 1428475026 T^{10} + 9515158664 T^{11} + 124800942246 T^{12} - 411307083359 T^{13} - 6208904645339 T^{14} + 198702541525 p T^{15} + 260553692815250 T^{16} + 198702541525 p^{2} T^{17} - 6208904645339 p^{2} T^{18} - 411307083359 p^{3} T^{19} + 124800942246 p^{4} T^{20} + 9515158664 p^{5} T^{21} - 1428475026 p^{6} T^{22} - 271175414 p^{7} T^{23} + 3448032 p^{8} T^{24} + 8402878 p^{9} T^{25} - 40953 p^{10} T^{26} - 176179 p^{11} T^{27} + 6137 p^{12} T^{28} + 2100 p^{13} T^{29} - 135 p^{14} T^{30} - 11 p^{15} T^{31} + p^{16} T^{32} \)
47 \( 1 - 5 T - 168 T^{2} + 161 T^{3} + 16599 T^{4} + 36967 T^{5} - 885940 T^{6} - 5198908 T^{7} + 21412749 T^{8} + 271776345 T^{9} + 346611318 T^{10} - 4913373621 T^{11} - 20631932251 T^{12} - 180472147091 T^{13} - 1523045035175 T^{14} + 7154378798768 T^{15} + 147263530000230 T^{16} + 7154378798768 p T^{17} - 1523045035175 p^{2} T^{18} - 180472147091 p^{3} T^{19} - 20631932251 p^{4} T^{20} - 4913373621 p^{5} T^{21} + 346611318 p^{6} T^{22} + 271776345 p^{7} T^{23} + 21412749 p^{8} T^{24} - 5198908 p^{9} T^{25} - 885940 p^{10} T^{26} + 36967 p^{11} T^{27} + 16599 p^{12} T^{28} + 161 p^{13} T^{29} - 168 p^{14} T^{30} - 5 p^{15} T^{31} + p^{16} T^{32} \)
53 \( 1 - 36 T + 508 T^{2} - 3932 T^{3} + 29444 T^{4} - 280817 T^{5} + 1836996 T^{6} - 8495258 T^{7} + 63588909 T^{8} - 353688724 T^{9} + 1151048267 T^{10} - 22337541825 T^{11} + 242039231570 T^{12} - 1659435705048 T^{13} + 17509974066650 T^{14} - 155428900722104 T^{15} + 1072759047046030 T^{16} - 155428900722104 p T^{17} + 17509974066650 p^{2} T^{18} - 1659435705048 p^{3} T^{19} + 242039231570 p^{4} T^{20} - 22337541825 p^{5} T^{21} + 1151048267 p^{6} T^{22} - 353688724 p^{7} T^{23} + 63588909 p^{8} T^{24} - 8495258 p^{9} T^{25} + 1836996 p^{10} T^{26} - 280817 p^{11} T^{27} + 29444 p^{12} T^{28} - 3932 p^{13} T^{29} + 508 p^{14} T^{30} - 36 p^{15} T^{31} + p^{16} T^{32} \)
59 \( ( 1 - 17 T + 406 T^{2} - 4290 T^{3} + 60383 T^{4} - 466365 T^{5} + 5192871 T^{6} - 33146017 T^{7} + 335346919 T^{8} - 33146017 p T^{9} + 5192871 p^{2} T^{10} - 466365 p^{3} T^{11} + 60383 p^{4} T^{12} - 4290 p^{5} T^{13} + 406 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
61 \( 1 + 22 T - 95 T^{2} - 3580 T^{3} + 31457 T^{4} + 560563 T^{5} - 4904245 T^{6} - 53746097 T^{7} + 634614646 T^{8} + 3868867413 T^{9} - 63304590509 T^{10} - 211847608419 T^{11} + 5064616862260 T^{12} + 7933710068297 T^{13} - 355599356859504 T^{14} - 168467878784829 T^{15} + 22415396390386199 T^{16} - 168467878784829 p T^{17} - 355599356859504 p^{2} T^{18} + 7933710068297 p^{3} T^{19} + 5064616862260 p^{4} T^{20} - 211847608419 p^{5} T^{21} - 63304590509 p^{6} T^{22} + 3868867413 p^{7} T^{23} + 634614646 p^{8} T^{24} - 53746097 p^{9} T^{25} - 4904245 p^{10} T^{26} + 560563 p^{11} T^{27} + 31457 p^{12} T^{28} - 3580 p^{13} T^{29} - 95 p^{14} T^{30} + 22 p^{15} T^{31} + p^{16} T^{32} \)
67 \( 1 - 26 T + 167 T^{2} + 144 T^{3} + 8449 T^{4} - 117319 T^{5} - 389361 T^{6} - 56323 p T^{7} + 147119946 T^{8} - 250628453 T^{9} + 228999159 T^{10} - 66354454115 T^{11} - 131549954032 T^{12} + 4100116909047 T^{13} + 36001458072338 T^{14} - 214087313935437 T^{15} - 1701481366216205 T^{16} - 214087313935437 p T^{17} + 36001458072338 p^{2} T^{18} + 4100116909047 p^{3} T^{19} - 131549954032 p^{4} T^{20} - 66354454115 p^{5} T^{21} + 228999159 p^{6} T^{22} - 250628453 p^{7} T^{23} + 147119946 p^{8} T^{24} - 56323 p^{10} T^{25} - 389361 p^{10} T^{26} - 117319 p^{11} T^{27} + 8449 p^{12} T^{28} + 144 p^{13} T^{29} + 167 p^{14} T^{30} - 26 p^{15} T^{31} + p^{16} T^{32} \)
71 \( 1 - 9 T - 386 T^{2} + 3589 T^{3} + 81235 T^{4} - 746618 T^{5} - 12406552 T^{6} + 106145741 T^{7} + 1537530601 T^{8} - 11338510824 T^{9} - 161456593527 T^{10} + 918377967383 T^{11} + 14822377342250 T^{12} - 52736724869388 T^{13} - 1218705028862170 T^{14} + 1440399689396096 T^{15} + 90831121962753112 T^{16} + 1440399689396096 p T^{17} - 1218705028862170 p^{2} T^{18} - 52736724869388 p^{3} T^{19} + 14822377342250 p^{4} T^{20} + 918377967383 p^{5} T^{21} - 161456593527 p^{6} T^{22} - 11338510824 p^{7} T^{23} + 1537530601 p^{8} T^{24} + 106145741 p^{9} T^{25} - 12406552 p^{10} T^{26} - 746618 p^{11} T^{27} + 81235 p^{12} T^{28} + 3589 p^{13} T^{29} - 386 p^{14} T^{30} - 9 p^{15} T^{31} + p^{16} T^{32} \)
73 \( 1 + 6 T - 165 T^{2} + 1806 T^{3} + 26272 T^{4} - 359064 T^{5} + 1422661 T^{6} + 45649262 T^{7} - 424402259 T^{8} + 411144950 T^{9} + 51687211764 T^{10} - 358428082760 T^{11} - 635781335617 T^{12} + 45102010283078 T^{13} - 217262895796719 T^{14} - 1212829928109410 T^{15} + 31435042908856916 T^{16} - 1212829928109410 p T^{17} - 217262895796719 p^{2} T^{18} + 45102010283078 p^{3} T^{19} - 635781335617 p^{4} T^{20} - 358428082760 p^{5} T^{21} + 51687211764 p^{6} T^{22} + 411144950 p^{7} T^{23} - 424402259 p^{8} T^{24} + 45649262 p^{9} T^{25} + 1422661 p^{10} T^{26} - 359064 p^{11} T^{27} + 26272 p^{12} T^{28} + 1806 p^{13} T^{29} - 165 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \)
79 \( 1 - 16 T - 231 T^{2} + 4320 T^{3} + 34115 T^{4} - 635696 T^{5} - 4129200 T^{6} + 64433872 T^{7} + 434574435 T^{8} - 5166633824 T^{9} - 32859288299 T^{10} + 304450300496 T^{11} + 1757236296274 T^{12} - 12357156907472 T^{13} - 63637790409375 T^{14} + 230917397502624 T^{15} + 3042650590829296 T^{16} + 230917397502624 p T^{17} - 63637790409375 p^{2} T^{18} - 12357156907472 p^{3} T^{19} + 1757236296274 p^{4} T^{20} + 304450300496 p^{5} T^{21} - 32859288299 p^{6} T^{22} - 5166633824 p^{7} T^{23} + 434574435 p^{8} T^{24} + 64433872 p^{9} T^{25} - 4129200 p^{10} T^{26} - 635696 p^{11} T^{27} + 34115 p^{12} T^{28} + 4320 p^{13} T^{29} - 231 p^{14} T^{30} - 16 p^{15} T^{31} + p^{16} T^{32} \)
83 \( ( 1 - 18 T + 468 T^{2} - 5152 T^{3} + 82570 T^{4} - 680729 T^{5} + 9262051 T^{6} - 65493403 T^{7} + 834546596 T^{8} - 65493403 p T^{9} + 9262051 p^{2} T^{10} - 680729 p^{3} T^{11} + 82570 p^{4} T^{12} - 5152 p^{5} T^{13} + 468 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
89 \( ( 1 + 20 T + 630 T^{2} + 9755 T^{3} + 178627 T^{4} + 2212339 T^{5} + 29910803 T^{6} + 302486077 T^{7} + 36604201 p T^{8} + 302486077 p T^{9} + 29910803 p^{2} T^{10} + 2212339 p^{3} T^{11} + 178627 p^{4} T^{12} + 9755 p^{5} T^{13} + 630 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
97 \( 1 - 7 T - 310 T^{2} + 2485 T^{3} + 42400 T^{4} - 379400 T^{5} - 3621348 T^{6} + 31983102 T^{7} + 207352950 T^{8} - 1459259835 T^{9} + 6146393957 T^{10} - 52607230120 T^{11} - 3573659402933 T^{12} + 20216002617763 T^{13} + 477664863434550 T^{14} - 1221840745845096 T^{15} - 46591681297798206 T^{16} - 1221840745845096 p T^{17} + 477664863434550 p^{2} T^{18} + 20216002617763 p^{3} T^{19} - 3573659402933 p^{4} T^{20} - 52607230120 p^{5} T^{21} + 6146393957 p^{6} T^{22} - 1459259835 p^{7} T^{23} + 207352950 p^{8} T^{24} + 31983102 p^{9} T^{25} - 3621348 p^{10} T^{26} - 379400 p^{11} T^{27} + 42400 p^{12} T^{28} + 2485 p^{13} T^{29} - 310 p^{14} T^{30} - 7 p^{15} T^{31} + p^{16} T^{32} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.47846302016209755015562231917, −3.38671145260066128958807052597, −3.33910074004508108901042267672, −3.21554651070689576493645361242, −3.19016661966284561660177921283, −2.89092541731920135437331883609, −2.77177600359856111131933405575, −2.73908988727750256857837265807, −2.66637623860664639029813389259, −2.49826537047105482002250463873, −2.43682975303337846775998107335, −2.42021904308716418993873679538, −2.21272095362163424881870186958, −2.20409486821433856676644571218, −2.16347565537627629671393030801, −2.12676436934481645113374256720, −1.89840338369393129563452682647, −1.62435351962758991586416749725, −1.48577486305277133139762548770, −1.22507305992390070665373344549, −1.05294394591876147595302837625, −1.01038734614467746215122824393, −0.798210055929101775812197716720, −0.59230488831685253035940714283, −0.59068248422909359583722099575, 0.59068248422909359583722099575, 0.59230488831685253035940714283, 0.798210055929101775812197716720, 1.01038734614467746215122824393, 1.05294394591876147595302837625, 1.22507305992390070665373344549, 1.48577486305277133139762548770, 1.62435351962758991586416749725, 1.89840338369393129563452682647, 2.12676436934481645113374256720, 2.16347565537627629671393030801, 2.20409486821433856676644571218, 2.21272095362163424881870186958, 2.42021904308716418993873679538, 2.43682975303337846775998107335, 2.49826537047105482002250463873, 2.66637623860664639029813389259, 2.73908988727750256857837265807, 2.77177600359856111131933405575, 2.89092541731920135437331883609, 3.19016661966284561660177921283, 3.21554651070689576493645361242, 3.33910074004508108901042267672, 3.38671145260066128958807052597, 3.47846302016209755015562231917

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.