Properties

Label 32-2736e16-1.1-c1e16-0-0
Degree $32$
Conductor $9.860\times 10^{54}$
Sign $1$
Analytic cond. $2.69334\times 10^{21}$
Root an. cond. $4.67408$
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  − 8·7-s − 24·13-s + 12·19-s − 10·25-s − 24·49-s − 44·61-s + 24·67-s − 20·73-s + 48·79-s + 192·91-s + 48·97-s + 132·109-s + 116·121-s + 127-s + 131-s − 96·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 228·169-s + 173-s + 80·175-s + 179-s + ⋯
L(s)  = 1  − 3.02·7-s − 6.65·13-s + 2.75·19-s − 2·25-s − 3.42·49-s − 5.63·61-s + 2.93·67-s − 2.34·73-s + 5.40·79-s + 20.1·91-s + 4.87·97-s + 12.6·109-s + 10.5·121-s + 0.0887·127-s + 0.0873·131-s − 8.32·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 17.5·169-s + 0.0760·173-s + 6.04·175-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{64} \cdot 3^{32} \cdot 19^{16}\)
Sign: $1$
Analytic conductor: \(2.69334\times 10^{21}\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{64} \cdot 3^{32} \cdot 19^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6302898464\)
\(L(\frac12)\) \(\approx\) \(0.6302898464\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( ( 1 - 6 T + 24 T^{2} + 6 p T^{3} - 662 T^{4} + 6 p^{2} T^{5} + 24 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
good5 \( 1 + 2 p T^{2} + 12 T^{4} - 348 T^{6} - 342 p T^{8} + 3102 T^{10} + 41864 T^{12} + 39446 T^{14} - 547929 T^{16} + 39446 p^{2} T^{18} + 41864 p^{4} T^{20} + 3102 p^{6} T^{22} - 342 p^{9} T^{24} - 348 p^{10} T^{26} + 12 p^{12} T^{28} + 2 p^{15} T^{30} + p^{16} T^{32} \)
7 \( ( 1 + 2 T + 16 T^{2} + 22 T^{3} + 137 T^{4} + 22 p T^{5} + 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
11 \( ( 1 - 58 T^{2} + 1648 T^{4} - 30062 T^{6} + 387698 T^{8} - 30062 p^{2} T^{10} + 1648 p^{4} T^{12} - 58 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
13 \( ( 1 + 12 T + 102 T^{2} + 648 T^{3} + 3449 T^{4} + 16236 T^{5} + 70182 T^{6} + 280320 T^{7} + 1051860 T^{8} + 280320 p T^{9} + 70182 p^{2} T^{10} + 16236 p^{3} T^{11} + 3449 p^{4} T^{12} + 648 p^{5} T^{13} + 102 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
17 \( 1 + 80 T^{2} + 3204 T^{4} + 87456 T^{6} + 1848714 T^{8} + 31251600 T^{10} + 432997904 T^{12} + 5576209648 T^{14} + 83165812371 T^{16} + 5576209648 p^{2} T^{18} + 432997904 p^{4} T^{20} + 31251600 p^{6} T^{22} + 1848714 p^{8} T^{24} + 87456 p^{10} T^{26} + 3204 p^{12} T^{28} + 80 p^{14} T^{30} + p^{16} T^{32} \)
23 \( 1 + 118 T^{2} + 7152 T^{4} + 293076 T^{6} + 9219354 T^{8} + 245859810 T^{10} + 6059768984 T^{12} + 145114787378 T^{14} + 3389582578647 T^{16} + 145114787378 p^{2} T^{18} + 6059768984 p^{4} T^{20} + 245859810 p^{6} T^{22} + 9219354 p^{8} T^{24} + 293076 p^{10} T^{26} + 7152 p^{12} T^{28} + 118 p^{14} T^{30} + p^{16} T^{32} \)
29 \( 1 - 168 T^{2} + 500 p T^{4} - 903600 T^{6} + 46201450 T^{8} - 2019949272 T^{10} + 77254987024 T^{12} - 2636561970840 T^{14} + 80765376191539 T^{16} - 2636561970840 p^{2} T^{18} + 77254987024 p^{4} T^{20} - 2019949272 p^{6} T^{22} + 46201450 p^{8} T^{24} - 903600 p^{10} T^{26} + 500 p^{13} T^{28} - 168 p^{14} T^{30} + p^{16} T^{32} \)
31 \( ( 1 - 168 T^{2} + 446 p T^{4} - 728100 T^{6} + 26741163 T^{8} - 728100 p^{2} T^{10} + 446 p^{5} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
37 \( ( 1 - 116 T^{2} + 7282 T^{4} - 309152 T^{6} + 11736187 T^{8} - 309152 p^{2} T^{10} + 7282 p^{4} T^{12} - 116 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
41 \( 1 - 144 T^{2} + 212 p T^{4} - 296352 T^{6} + 8730346 T^{8} - 423855504 T^{10} + 390696656 p T^{12} - 49180899888 T^{14} - 14515835440685 T^{16} - 49180899888 p^{2} T^{18} + 390696656 p^{5} T^{20} - 423855504 p^{6} T^{22} + 8730346 p^{8} T^{24} - 296352 p^{10} T^{26} + 212 p^{13} T^{28} - 144 p^{14} T^{30} + p^{16} T^{32} \)
43 \( ( 1 - 128 T^{2} + 84 T^{3} + 8875 T^{4} - 7182 T^{5} - 486044 T^{6} + 146958 T^{7} + 22596412 T^{8} + 146958 p T^{9} - 486044 p^{2} T^{10} - 7182 p^{3} T^{11} + 8875 p^{4} T^{12} + 84 p^{5} T^{13} - 128 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
47 \( 1 + 64 T^{2} - 5124 T^{4} - 235392 T^{6} + 27710346 T^{8} + 764142144 T^{10} - 82759440400 T^{12} - 496594517824 T^{14} + 227129400113043 T^{16} - 496594517824 p^{2} T^{18} - 82759440400 p^{4} T^{20} + 764142144 p^{6} T^{22} + 27710346 p^{8} T^{24} - 235392 p^{10} T^{26} - 5124 p^{12} T^{28} + 64 p^{14} T^{30} + p^{16} T^{32} \)
53 \( 1 - 334 T^{2} + 59364 T^{4} - 7457900 T^{6} + 739217570 T^{8} - 61143323490 T^{10} + 82301828120 p T^{12} - 274000136012722 T^{14} + 15344873339596263 T^{16} - 274000136012722 p^{2} T^{18} + 82301828120 p^{5} T^{20} - 61143323490 p^{6} T^{22} + 739217570 p^{8} T^{24} - 7457900 p^{10} T^{26} + 59364 p^{12} T^{28} - 334 p^{14} T^{30} + p^{16} T^{32} \)
59 \( 1 - 366 T^{2} + 70420 T^{4} - 9710316 T^{6} + 1075327138 T^{8} - 100300555314 T^{10} + 8085095078104 T^{12} - 572375369123922 T^{14} + 35862697927173799 T^{16} - 572375369123922 p^{2} T^{18} + 8085095078104 p^{4} T^{20} - 100300555314 p^{6} T^{22} + 1075327138 p^{8} T^{24} - 9710316 p^{10} T^{26} + 70420 p^{12} T^{28} - 366 p^{14} T^{30} + p^{16} T^{32} \)
61 \( ( 1 + 22 T + 156 T^{2} + 696 T^{3} + 7635 T^{4} + 26064 T^{5} - 565456 T^{6} - 6581302 T^{7} - 45855456 T^{8} - 6581302 p T^{9} - 565456 p^{2} T^{10} + 26064 p^{3} T^{11} + 7635 p^{4} T^{12} + 696 p^{5} T^{13} + 156 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
67 \( ( 1 - 12 T + 138 T^{2} - 1080 T^{3} + 10157 T^{4} - 30774 T^{5} - 273306 T^{6} + 4048566 T^{7} - 34546128 T^{8} + 4048566 p T^{9} - 273306 p^{2} T^{10} - 30774 p^{3} T^{11} + 10157 p^{4} T^{12} - 1080 p^{5} T^{13} + 138 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
71 \( 1 - 208 T^{2} + 21564 T^{4} - 1548896 T^{6} + 71833802 T^{8} - 1897150320 T^{10} - 3138432112 p T^{12} + 48475735509968 T^{14} - 4373142692439213 T^{16} + 48475735509968 p^{2} T^{18} - 3138432112 p^{5} T^{20} - 1897150320 p^{6} T^{22} + 71833802 p^{8} T^{24} - 1548896 p^{10} T^{26} + 21564 p^{12} T^{28} - 208 p^{14} T^{30} + p^{16} T^{32} \)
73 \( ( 1 + 10 T + 20 T^{2} + 248 T^{3} + 1151 T^{4} - 7552 T^{5} + 528216 T^{6} + 5616654 T^{7} + 15933832 T^{8} + 5616654 p T^{9} + 528216 p^{2} T^{10} - 7552 p^{3} T^{11} + 1151 p^{4} T^{12} + 248 p^{5} T^{13} + 20 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
79 \( ( 1 - 24 T + 540 T^{2} - 8352 T^{3} + 122339 T^{4} - 1457442 T^{5} + 16670928 T^{6} - 164188794 T^{7} + 1562088492 T^{8} - 164188794 p T^{9} + 16670928 p^{2} T^{10} - 1457442 p^{3} T^{11} + 122339 p^{4} T^{12} - 8352 p^{5} T^{13} + 540 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
83 \( ( 1 - 208 T^{2} + 16060 T^{4} - 1135760 T^{6} + 110835494 T^{8} - 1135760 p^{2} T^{10} + 16060 p^{4} T^{12} - 208 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
89 \( 1 - 274 T^{2} + 36744 T^{4} - 2299580 T^{6} - 44111350 T^{8} + 23779348482 T^{10} - 2201904716792 T^{12} + 97647219822842 T^{14} - 2985825074618025 T^{16} + 97647219822842 p^{2} T^{18} - 2201904716792 p^{4} T^{20} + 23779348482 p^{6} T^{22} - 44111350 p^{8} T^{24} - 2299580 p^{10} T^{26} + 36744 p^{12} T^{28} - 274 p^{14} T^{30} + p^{16} T^{32} \)
97 \( ( 1 - 24 T + 504 T^{2} - 7488 T^{3} + 95462 T^{4} - 1096500 T^{5} + 12157968 T^{6} - 125319456 T^{7} + 1307837715 T^{8} - 125319456 p T^{9} + 12157968 p^{2} T^{10} - 1096500 p^{3} T^{11} + 95462 p^{4} T^{12} - 7488 p^{5} T^{13} + 504 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
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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

−2.10862822470127476222596269197, −2.09147314444021357922467573050, −2.08601173053292044635137622836, −2.01065874367336000414782692289, −1.94189862981991518698889226392, −1.80755041444206623551251189586, −1.80038987382010449481743592560, −1.75003799099346744756082748895, −1.66927731780250024668589390444, −1.60549854160773901247842049490, −1.49685742318429634611917934522, −1.35744811628746641401132871095, −1.18548958650470882389318546987, −1.12167826097870157752291231024, −1.10984711306556239010798920903, −0.933966506999798419273662369624, −0.811870812151986389914248621657, −0.76941398849144673047437918347, −0.73987037631247448661486609133, −0.45712187904732907422998685591, −0.44386397145313041302127477723, −0.34304620512433182976511693944, −0.31129780536056276204859173882, −0.14415779199146435884954365668, −0.11574540033293784078162547622, 0.11574540033293784078162547622, 0.14415779199146435884954365668, 0.31129780536056276204859173882, 0.34304620512433182976511693944, 0.44386397145313041302127477723, 0.45712187904732907422998685591, 0.73987037631247448661486609133, 0.76941398849144673047437918347, 0.811870812151986389914248621657, 0.933966506999798419273662369624, 1.10984711306556239010798920903, 1.12167826097870157752291231024, 1.18548958650470882389318546987, 1.35744811628746641401132871095, 1.49685742318429634611917934522, 1.60549854160773901247842049490, 1.66927731780250024668589390444, 1.75003799099346744756082748895, 1.80038987382010449481743592560, 1.80755041444206623551251189586, 1.94189862981991518698889226392, 2.01065874367336000414782692289, 2.08601173053292044635137622836, 2.09147314444021357922467573050, 2.10862822470127476222596269197

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

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