Properties

Label 28-6018e14-1.1-c1e14-0-0
Degree $28$
Conductor $8.172\times 10^{52}$
Sign $1$
Analytic cond. $3.50114\times 10^{23}$
Root an. cond. $6.93209$
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  + 14·2-s − 14·3-s + 105·4-s + 2·5-s − 196·6-s + 7-s + 560·8-s + 105·9-s + 28·10-s + 3·11-s − 1.47e3·12-s + 16·13-s + 14·14-s − 28·15-s + 2.38e3·16-s + 14·17-s + 1.47e3·18-s + 13·19-s + 210·20-s − 14·21-s + 42·22-s + 4·23-s − 7.84e3·24-s − 17·25-s + 224·26-s − 560·27-s + 105·28-s + ⋯
L(s)  = 1  + 9.89·2-s − 8.08·3-s + 52.5·4-s + 0.894·5-s − 80.0·6-s + 0.377·7-s + 197.·8-s + 35·9-s + 8.85·10-s + 0.904·11-s − 424.·12-s + 4.43·13-s + 3.74·14-s − 7.22·15-s + 595·16-s + 3.39·17-s + 346.·18-s + 2.98·19-s + 46.9·20-s − 3.05·21-s + 8.95·22-s + 0.834·23-s − 1.60e3·24-s − 3.39·25-s + 43.9·26-s − 107.·27-s + 19.8·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 3^{14} \cdot 17^{14} \cdot 59^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 3^{14} \cdot 17^{14} \cdot 59^{14}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(28\)
Conductor: \(2^{14} \cdot 3^{14} \cdot 17^{14} \cdot 59^{14}\)
Sign: $1$
Analytic conductor: \(3.50114\times 10^{23}\)
Root analytic conductor: \(6.93209\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((28,\ 2^{14} \cdot 3^{14} \cdot 17^{14} \cdot 59^{14} ,\ ( \ : [1/2]^{14} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(180293.8330\)
\(L(\frac12)\) \(\approx\) \(180293.8330\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T )^{14} \)
3 \( ( 1 + T )^{14} \)
17 \( ( 1 - T )^{14} \)
59 \( ( 1 - T )^{14} \)
good5 \( 1 - 2 T + 21 T^{2} - 51 T^{3} + 291 T^{4} - 734 T^{5} + 3054 T^{6} - 1523 p T^{7} + 5271 p T^{8} - 62574 T^{9} + 191183 T^{10} - 426649 T^{11} + 1189881 T^{12} - 2481267 T^{13} + 6385916 T^{14} - 2481267 p T^{15} + 1189881 p^{2} T^{16} - 426649 p^{3} T^{17} + 191183 p^{4} T^{18} - 62574 p^{5} T^{19} + 5271 p^{7} T^{20} - 1523 p^{8} T^{21} + 3054 p^{8} T^{22} - 734 p^{9} T^{23} + 291 p^{10} T^{24} - 51 p^{11} T^{25} + 21 p^{12} T^{26} - 2 p^{13} T^{27} + p^{14} T^{28} \)
7 \( 1 - T + 25 T^{2} - 25 T^{3} + 325 T^{4} - 118 T^{5} + 3065 T^{6} + 821 T^{7} + 25755 T^{8} + 16424 T^{9} + 216278 T^{10} + 194154 T^{11} + 230697 p T^{12} + 1881109 T^{13} + 11176912 T^{14} + 1881109 p T^{15} + 230697 p^{3} T^{16} + 194154 p^{3} T^{17} + 216278 p^{4} T^{18} + 16424 p^{5} T^{19} + 25755 p^{6} T^{20} + 821 p^{7} T^{21} + 3065 p^{8} T^{22} - 118 p^{9} T^{23} + 325 p^{10} T^{24} - 25 p^{11} T^{25} + 25 p^{12} T^{26} - p^{13} T^{27} + p^{14} T^{28} \)
11 \( 1 - 3 T + 50 T^{2} - 139 T^{3} + 1389 T^{4} - 3461 T^{5} + 26271 T^{6} - 55628 T^{7} + 382329 T^{8} - 644388 T^{9} + 4553430 T^{10} - 5828948 T^{11} + 48935049 T^{12} - 49927459 T^{13} + 521617234 T^{14} - 49927459 p T^{15} + 48935049 p^{2} T^{16} - 5828948 p^{3} T^{17} + 4553430 p^{4} T^{18} - 644388 p^{5} T^{19} + 382329 p^{6} T^{20} - 55628 p^{7} T^{21} + 26271 p^{8} T^{22} - 3461 p^{9} T^{23} + 1389 p^{10} T^{24} - 139 p^{11} T^{25} + 50 p^{12} T^{26} - 3 p^{13} T^{27} + p^{14} T^{28} \)
13 \( 1 - 16 T + p^{2} T^{2} - 1546 T^{3} + 12022 T^{4} - 82182 T^{5} + 516052 T^{6} - 2968622 T^{7} + 15781456 T^{8} - 78678738 T^{9} + 28261163 p T^{10} - 1611699964 T^{11} + 6689981345 T^{12} - 26213197492 T^{13} + 97044671368 T^{14} - 26213197492 p T^{15} + 6689981345 p^{2} T^{16} - 1611699964 p^{3} T^{17} + 28261163 p^{5} T^{18} - 78678738 p^{5} T^{19} + 15781456 p^{6} T^{20} - 2968622 p^{7} T^{21} + 516052 p^{8} T^{22} - 82182 p^{9} T^{23} + 12022 p^{10} T^{24} - 1546 p^{11} T^{25} + p^{14} T^{26} - 16 p^{13} T^{27} + p^{14} T^{28} \)
19 \( 1 - 13 T + 113 T^{2} - 600 T^{3} + 2592 T^{4} - 9014 T^{5} + 46757 T^{6} - 15439 p T^{7} + 1849500 T^{8} - 8966661 T^{9} + 37800767 T^{10} - 148231206 T^{11} + 680287275 T^{12} - 176814149 p T^{13} + 15854162454 T^{14} - 176814149 p^{2} T^{15} + 680287275 p^{2} T^{16} - 148231206 p^{3} T^{17} + 37800767 p^{4} T^{18} - 8966661 p^{5} T^{19} + 1849500 p^{6} T^{20} - 15439 p^{8} T^{21} + 46757 p^{8} T^{22} - 9014 p^{9} T^{23} + 2592 p^{10} T^{24} - 600 p^{11} T^{25} + 113 p^{12} T^{26} - 13 p^{13} T^{27} + p^{14} T^{28} \)
23 \( 1 - 4 T + 154 T^{2} - 568 T^{3} + 12794 T^{4} - 44079 T^{5} + 741240 T^{6} - 2389906 T^{7} + 33104013 T^{8} - 99791439 T^{9} + 51921141 p T^{10} - 3352367173 T^{11} + 35677970512 T^{12} - 92717378581 T^{13} + 894106073478 T^{14} - 92717378581 p T^{15} + 35677970512 p^{2} T^{16} - 3352367173 p^{3} T^{17} + 51921141 p^{5} T^{18} - 99791439 p^{5} T^{19} + 33104013 p^{6} T^{20} - 2389906 p^{7} T^{21} + 741240 p^{8} T^{22} - 44079 p^{9} T^{23} + 12794 p^{10} T^{24} - 568 p^{11} T^{25} + 154 p^{12} T^{26} - 4 p^{13} T^{27} + p^{14} T^{28} \)
29 \( 1 + 165 T^{2} + 189 T^{3} + 15363 T^{4} + 28744 T^{5} + 1034374 T^{6} + 2331285 T^{7} + 55569405 T^{8} + 132007708 T^{9} + 2456429475 T^{10} + 5819632451 T^{11} + 91016171919 T^{12} + 207253218643 T^{13} + 2858058200868 T^{14} + 207253218643 p T^{15} + 91016171919 p^{2} T^{16} + 5819632451 p^{3} T^{17} + 2456429475 p^{4} T^{18} + 132007708 p^{5} T^{19} + 55569405 p^{6} T^{20} + 2331285 p^{7} T^{21} + 1034374 p^{8} T^{22} + 28744 p^{9} T^{23} + 15363 p^{10} T^{24} + 189 p^{11} T^{25} + 165 p^{12} T^{26} + p^{14} T^{28} \)
31 \( 1 + 13 T + 312 T^{2} + 3067 T^{3} + 43752 T^{4} + 11458 p T^{5} + 3876594 T^{6} + 27323974 T^{7} + 250569130 T^{8} + 1577954975 T^{9} + 13216560 p^{2} T^{10} + 72496609665 T^{11} + 523292575149 T^{12} + 2721705970904 T^{13} + 17808334356812 T^{14} + 2721705970904 p T^{15} + 523292575149 p^{2} T^{16} + 72496609665 p^{3} T^{17} + 13216560 p^{6} T^{18} + 1577954975 p^{5} T^{19} + 250569130 p^{6} T^{20} + 27323974 p^{7} T^{21} + 3876594 p^{8} T^{22} + 11458 p^{10} T^{23} + 43752 p^{10} T^{24} + 3067 p^{11} T^{25} + 312 p^{12} T^{26} + 13 p^{13} T^{27} + p^{14} T^{28} \)
37 \( 1 - 12 T + 423 T^{2} - 4103 T^{3} + 82191 T^{4} - 675686 T^{5} + 9987728 T^{6} - 71557065 T^{7} + 861113293 T^{8} - 5468136484 T^{9} + 56258953649 T^{10} - 8642628049 p T^{11} + 2890375230979 T^{12} - 14770249145913 T^{13} + 119067609810896 T^{14} - 14770249145913 p T^{15} + 2890375230979 p^{2} T^{16} - 8642628049 p^{4} T^{17} + 56258953649 p^{4} T^{18} - 5468136484 p^{5} T^{19} + 861113293 p^{6} T^{20} - 71557065 p^{7} T^{21} + 9987728 p^{8} T^{22} - 675686 p^{9} T^{23} + 82191 p^{10} T^{24} - 4103 p^{11} T^{25} + 423 p^{12} T^{26} - 12 p^{13} T^{27} + p^{14} T^{28} \)
41 \( 1 + 18 T + 300 T^{2} + 3524 T^{3} + 38060 T^{4} + 349017 T^{5} + 3032646 T^{6} + 23315036 T^{7} + 173395621 T^{8} + 1165742335 T^{9} + 7708348011 T^{10} + 47218990227 T^{11} + 294302573194 T^{12} + 1756965391355 T^{13} + 11341880330478 T^{14} + 1756965391355 p T^{15} + 294302573194 p^{2} T^{16} + 47218990227 p^{3} T^{17} + 7708348011 p^{4} T^{18} + 1165742335 p^{5} T^{19} + 173395621 p^{6} T^{20} + 23315036 p^{7} T^{21} + 3032646 p^{8} T^{22} + 349017 p^{9} T^{23} + 38060 p^{10} T^{24} + 3524 p^{11} T^{25} + 300 p^{12} T^{26} + 18 p^{13} T^{27} + p^{14} T^{28} \)
43 \( 1 - 29 T + 678 T^{2} - 11441 T^{3} + 169551 T^{4} - 2150511 T^{5} + 24865455 T^{6} - 259971598 T^{7} + 2526123999 T^{8} - 22752586230 T^{9} + 192590919626 T^{10} - 1530629632026 T^{11} + 11505849212241 T^{12} - 81734080633443 T^{13} + 550792279288290 T^{14} - 81734080633443 p T^{15} + 11505849212241 p^{2} T^{16} - 1530629632026 p^{3} T^{17} + 192590919626 p^{4} T^{18} - 22752586230 p^{5} T^{19} + 2526123999 p^{6} T^{20} - 259971598 p^{7} T^{21} + 24865455 p^{8} T^{22} - 2150511 p^{9} T^{23} + 169551 p^{10} T^{24} - 11441 p^{11} T^{25} + 678 p^{12} T^{26} - 29 p^{13} T^{27} + p^{14} T^{28} \)
47 \( 1 + 320 T^{2} + 301 T^{3} + 51002 T^{4} + 87475 T^{5} + 5567082 T^{6} + 12809341 T^{7} + 471115484 T^{8} + 1278265771 T^{9} + 32681023288 T^{10} + 96120182430 T^{11} + 1917619523337 T^{12} + 5661492070954 T^{13} + 96874224484668 T^{14} + 5661492070954 p T^{15} + 1917619523337 p^{2} T^{16} + 96120182430 p^{3} T^{17} + 32681023288 p^{4} T^{18} + 1278265771 p^{5} T^{19} + 471115484 p^{6} T^{20} + 12809341 p^{7} T^{21} + 5567082 p^{8} T^{22} + 87475 p^{9} T^{23} + 51002 p^{10} T^{24} + 301 p^{11} T^{25} + 320 p^{12} T^{26} + p^{14} T^{28} \)
53 \( 1 - 24 T + 361 T^{2} - 4358 T^{3} + 51763 T^{4} - 581295 T^{5} + 6018642 T^{6} - 57193858 T^{7} + 523629997 T^{8} - 4600472695 T^{9} + 39080179363 T^{10} - 317550307875 T^{11} + 2503130140719 T^{12} - 19011879973971 T^{13} + 140649127992036 T^{14} - 19011879973971 p T^{15} + 2503130140719 p^{2} T^{16} - 317550307875 p^{3} T^{17} + 39080179363 p^{4} T^{18} - 4600472695 p^{5} T^{19} + 523629997 p^{6} T^{20} - 57193858 p^{7} T^{21} + 6018642 p^{8} T^{22} - 581295 p^{9} T^{23} + 51763 p^{10} T^{24} - 4358 p^{11} T^{25} + 361 p^{12} T^{26} - 24 p^{13} T^{27} + p^{14} T^{28} \)
61 \( 1 - 29 T + 686 T^{2} - 11561 T^{3} + 177227 T^{4} - 2304239 T^{5} + 28369001 T^{6} - 315196332 T^{7} + 3376423357 T^{8} - 33485617158 T^{9} + 322132315586 T^{10} - 2900245110860 T^{11} + 25395377224503 T^{12} - 209419131711817 T^{13} + 1683432433309326 T^{14} - 209419131711817 p T^{15} + 25395377224503 p^{2} T^{16} - 2900245110860 p^{3} T^{17} + 322132315586 p^{4} T^{18} - 33485617158 p^{5} T^{19} + 3376423357 p^{6} T^{20} - 315196332 p^{7} T^{21} + 28369001 p^{8} T^{22} - 2304239 p^{9} T^{23} + 177227 p^{10} T^{24} - 11561 p^{11} T^{25} + 686 p^{12} T^{26} - 29 p^{13} T^{27} + p^{14} T^{28} \)
67 \( 1 - 4 T + 498 T^{2} - 1605 T^{3} + 129226 T^{4} - 326903 T^{5} + 22689919 T^{6} - 43351466 T^{7} + 2997240462 T^{8} - 4182976544 T^{9} + 315355718058 T^{10} - 320339395387 T^{11} + 27347530935951 T^{12} - 21755614250359 T^{13} + 1992662222964298 T^{14} - 21755614250359 p T^{15} + 27347530935951 p^{2} T^{16} - 320339395387 p^{3} T^{17} + 315355718058 p^{4} T^{18} - 4182976544 p^{5} T^{19} + 2997240462 p^{6} T^{20} - 43351466 p^{7} T^{21} + 22689919 p^{8} T^{22} - 326903 p^{9} T^{23} + 129226 p^{10} T^{24} - 1605 p^{11} T^{25} + 498 p^{12} T^{26} - 4 p^{13} T^{27} + p^{14} T^{28} \)
71 \( 1 + 10 T + 666 T^{2} + 6631 T^{3} + 225458 T^{4} + 2148131 T^{5} + 50456890 T^{6} + 450471985 T^{7} + 8248681276 T^{8} + 68128053057 T^{9} + 1035356484270 T^{10} + 7833433277004 T^{11} + 102526213872385 T^{12} + 703607693868982 T^{13} + 8121497837103868 T^{14} + 703607693868982 p T^{15} + 102526213872385 p^{2} T^{16} + 7833433277004 p^{3} T^{17} + 1035356484270 p^{4} T^{18} + 68128053057 p^{5} T^{19} + 8248681276 p^{6} T^{20} + 450471985 p^{7} T^{21} + 50456890 p^{8} T^{22} + 2148131 p^{9} T^{23} + 225458 p^{10} T^{24} + 6631 p^{11} T^{25} + 666 p^{12} T^{26} + 10 p^{13} T^{27} + p^{14} T^{28} \)
73 \( 1 - 18 T + 672 T^{2} - 10062 T^{3} + 222150 T^{4} - 2848505 T^{5} + 47679362 T^{6} - 536813156 T^{7} + 7463244543 T^{8} - 75063208337 T^{9} + 906770801297 T^{10} - 8238509429507 T^{11} + 88687914675302 T^{12} - 732167924532055 T^{13} + 7122079484889698 T^{14} - 732167924532055 p T^{15} + 88687914675302 p^{2} T^{16} - 8238509429507 p^{3} T^{17} + 906770801297 p^{4} T^{18} - 75063208337 p^{5} T^{19} + 7463244543 p^{6} T^{20} - 536813156 p^{7} T^{21} + 47679362 p^{8} T^{22} - 2848505 p^{9} T^{23} + 222150 p^{10} T^{24} - 10062 p^{11} T^{25} + 672 p^{12} T^{26} - 18 p^{13} T^{27} + p^{14} T^{28} \)
79 \( 1 - 7 T + 674 T^{2} - 4570 T^{3} + 227542 T^{4} - 1467070 T^{5} + 50890930 T^{6} - 308971891 T^{7} + 8432570604 T^{8} - 605943375 p T^{9} + 1096851691922 T^{10} - 5781333156354 T^{11} + 115625508535133 T^{12} - 561004984972953 T^{13} + 10039728020994228 T^{14} - 561004984972953 p T^{15} + 115625508535133 p^{2} T^{16} - 5781333156354 p^{3} T^{17} + 1096851691922 p^{4} T^{18} - 605943375 p^{6} T^{19} + 8432570604 p^{6} T^{20} - 308971891 p^{7} T^{21} + 50890930 p^{8} T^{22} - 1467070 p^{9} T^{23} + 227542 p^{10} T^{24} - 4570 p^{11} T^{25} + 674 p^{12} T^{26} - 7 p^{13} T^{27} + p^{14} T^{28} \)
83 \( 1 - 28 T + 814 T^{2} - 15360 T^{3} + 287657 T^{4} - 4329789 T^{5} + 63959827 T^{6} - 819211258 T^{7} + 10319080017 T^{8} - 116987636283 T^{9} + 1310083792898 T^{10} - 13477931766679 T^{11} + 137587311337173 T^{12} - 1304884343012113 T^{13} + 12312656757204954 T^{14} - 1304884343012113 p T^{15} + 137587311337173 p^{2} T^{16} - 13477931766679 p^{3} T^{17} + 1310083792898 p^{4} T^{18} - 116987636283 p^{5} T^{19} + 10319080017 p^{6} T^{20} - 819211258 p^{7} T^{21} + 63959827 p^{8} T^{22} - 4329789 p^{9} T^{23} + 287657 p^{10} T^{24} - 15360 p^{11} T^{25} + 814 p^{12} T^{26} - 28 p^{13} T^{27} + p^{14} T^{28} \)
89 \( 1 - 23 T + 1110 T^{2} - 20566 T^{3} + 565071 T^{4} - 8808956 T^{5} + 178161126 T^{6} - 2400867394 T^{7} + 39325272075 T^{8} - 465823934453 T^{9} + 6475794547973 T^{10} - 68066129273504 T^{11} + 823933553281769 T^{12} - 7712886392947376 T^{13} + 82456312607920630 T^{14} - 7712886392947376 p T^{15} + 823933553281769 p^{2} T^{16} - 68066129273504 p^{3} T^{17} + 6475794547973 p^{4} T^{18} - 465823934453 p^{5} T^{19} + 39325272075 p^{6} T^{20} - 2400867394 p^{7} T^{21} + 178161126 p^{8} T^{22} - 8808956 p^{9} T^{23} + 565071 p^{10} T^{24} - 20566 p^{11} T^{25} + 1110 p^{12} T^{26} - 23 p^{13} T^{27} + p^{14} T^{28} \)
97 \( 1 + 7 T + 770 T^{2} + 5617 T^{3} + 303931 T^{4} + 2206674 T^{5} + 80822725 T^{6} + 568381591 T^{7} + 16098182607 T^{8} + 107708662690 T^{9} + 2535410410950 T^{10} + 15918672408650 T^{11} + 325492809346837 T^{12} + 1892769552034803 T^{13} + 34607254064477606 T^{14} + 1892769552034803 p T^{15} + 325492809346837 p^{2} T^{16} + 15918672408650 p^{3} T^{17} + 2535410410950 p^{4} T^{18} + 107708662690 p^{5} T^{19} + 16098182607 p^{6} T^{20} + 568381591 p^{7} T^{21} + 80822725 p^{8} T^{22} + 2206674 p^{9} T^{23} + 303931 p^{10} T^{24} + 5617 p^{11} T^{25} + 770 p^{12} T^{26} + 7 p^{13} T^{27} + p^{14} T^{28} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.01898313804184787912121790401, −1.97725464477876745243594003818, −1.91989328412706453867136677060, −1.91539510998333472681953843450, −1.86285326087775444626233379169, −1.85183814747660507713103028422, −1.81323836647756392802692023626, −1.77536196816128511974458573226, −1.76638079464034342715751139484, −1.55532273585185139520778871310, −1.53265183867394003651292607715, −1.33554045646972392908771328542, −1.13708404011682069249728921373, −1.12000891767959020978895166499, −1.07940386413899735681110138458, −0.997419449669069769123660119227, −0.971771713180815008193905189964, −0.904262550273856531621613818251, −0.806869583759140423853089603070, −0.70332761102853930559415672841, −0.68610973326592892936738585652, −0.63335502857177389372595658008, −0.61748301049452451344573315812, −0.40318287333234964180493139396, −0.32182456169688634505070105854, 0.32182456169688634505070105854, 0.40318287333234964180493139396, 0.61748301049452451344573315812, 0.63335502857177389372595658008, 0.68610973326592892936738585652, 0.70332761102853930559415672841, 0.806869583759140423853089603070, 0.904262550273856531621613818251, 0.971771713180815008193905189964, 0.997419449669069769123660119227, 1.07940386413899735681110138458, 1.12000891767959020978895166499, 1.13708404011682069249728921373, 1.33554045646972392908771328542, 1.53265183867394003651292607715, 1.55532273585185139520778871310, 1.76638079464034342715751139484, 1.77536196816128511974458573226, 1.81323836647756392802692023626, 1.85183814747660507713103028422, 1.86285326087775444626233379169, 1.91539510998333472681953843450, 1.91989328412706453867136677060, 1.97725464477876745243594003818, 2.01898313804184787912121790401

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

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