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Properties

Label 12-945e6-1.1-c1e6-0-0
Degree $12$
Conductor $7.122\times 10^{17}$
Sign $1$
Analytic cond. $184609.$
Root an. cond. $2.74697$
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  + 3·2-s + 2·4-s + 3·5-s − 4·7-s − 3·8-s + 9·10-s − 9·11-s − 12·14-s − 4·16-s − 3·17-s + 21·19-s + 6·20-s − 27·22-s − 6·23-s + 3·25-s − 8·28-s + 6·31-s + 6·32-s − 9·34-s − 12·35-s + 16·37-s + 63·38-s − 9·40-s − 24·41-s − 8·43-s − 18·44-s − 18·46-s + ⋯
L(s)  = 1  + 2.12·2-s + 4-s + 1.34·5-s − 1.51·7-s − 1.06·8-s + 2.84·10-s − 2.71·11-s − 3.20·14-s − 16-s − 0.727·17-s + 4.81·19-s + 1.34·20-s − 5.75·22-s − 1.25·23-s + 3/5·25-s − 1.51·28-s + 1.07·31-s + 1.06·32-s − 1.54·34-s − 2.02·35-s + 2.63·37-s + 10.2·38-s − 1.42·40-s − 3.74·41-s − 1.21·43-s − 2.71·44-s − 2.65·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{18} \cdot 5^{6} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(184609.\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{18} \cdot 5^{6} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.7101228367\)
\(L(\frac12)\) \(\approx\) \(0.7101228367\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( ( 1 - T + T^{2} )^{3} \)
7 \( 1 + 4 T + 2 p T^{2} + 55 T^{3} + 2 p^{2} T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
good2 \( 1 - 3 T + 7 T^{2} - 3 p^{2} T^{3} + 17 T^{4} - 3 p^{3} T^{5} + 31 T^{6} - 3 p^{4} T^{7} + 17 p^{2} T^{8} - 3 p^{5} T^{9} + 7 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 9 T + 58 T^{2} + 279 T^{3} + 1151 T^{4} + 4398 T^{5} + 15085 T^{6} + 4398 p T^{7} + 1151 p^{2} T^{8} + 279 p^{3} T^{9} + 58 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 48 T^{2} + 1164 T^{4} - 18191 T^{6} + 1164 p^{2} T^{8} - 48 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 + 3 T - 9 T^{2} - 96 T^{3} - 207 T^{4} + 381 T^{5} + 7342 T^{6} + 381 p T^{7} - 207 p^{2} T^{8} - 96 p^{3} T^{9} - 9 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 21 T + 240 T^{2} - 1953 T^{3} + 12657 T^{4} - 69144 T^{5} + 324421 T^{6} - 69144 p T^{7} + 12657 p^{2} T^{8} - 1953 p^{3} T^{9} + 240 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 6 T + 82 T^{2} + 420 T^{3} + 3836 T^{4} + 15936 T^{5} + 106861 T^{6} + 15936 p T^{7} + 3836 p^{2} T^{8} + 420 p^{3} T^{9} + 82 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 25 T^{2} - 259 T^{4} - 45203 T^{6} - 259 p^{2} T^{8} + 25 p^{4} T^{10} + p^{6} T^{12} \)
31 \( 1 - 6 T + 90 T^{2} - 468 T^{3} + 4248 T^{4} - 16932 T^{5} + 145717 T^{6} - 16932 p T^{7} + 4248 p^{2} T^{8} - 468 p^{3} T^{9} + 90 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 16 T + 110 T^{2} - 426 T^{3} + 634 T^{4} + 11530 T^{5} - 125825 T^{6} + 11530 p T^{7} + 634 p^{2} T^{8} - 426 p^{3} T^{9} + 110 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 + 12 T + 162 T^{2} + 1011 T^{3} + 162 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 + 4 T + 65 T^{2} + 280 T^{3} + 65 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 + 21 T + 162 T^{2} + 1353 T^{3} + 17415 T^{4} + 124230 T^{5} + 650959 T^{6} + 124230 p T^{7} + 17415 p^{2} T^{8} + 1353 p^{3} T^{9} + 162 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 27 T + 460 T^{2} + 5859 T^{3} + 61103 T^{4} + 539352 T^{5} + 4194451 T^{6} + 539352 p T^{7} + 61103 p^{2} T^{8} + 5859 p^{3} T^{9} + 460 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 3 T - 60 T^{2} - 987 T^{3} - 1101 T^{4} + 27210 T^{5} + 454939 T^{6} + 27210 p T^{7} - 1101 p^{2} T^{8} - 987 p^{3} T^{9} - 60 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 33 T + 606 T^{2} + 8019 T^{3} + 82671 T^{4} + 726492 T^{5} + 5868979 T^{6} + 726492 p T^{7} + 82671 p^{2} T^{8} + 8019 p^{3} T^{9} + 606 p^{4} T^{10} + 33 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 27 T + 6 p T^{2} + 3215 T^{3} + 9231 T^{4} - 143994 T^{5} - 1955949 T^{6} - 143994 p T^{7} + 9231 p^{2} T^{8} + 3215 p^{3} T^{9} + 6 p^{5} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 98 T^{2} + 17423 T^{4} - 1003580 T^{6} + 17423 p^{2} T^{8} - 98 p^{4} T^{10} + p^{6} T^{12} \)
73 \( 1 + 12 T + 246 T^{2} + 2376 T^{3} + 31722 T^{4} + 281166 T^{5} + 2840749 T^{6} + 281166 p T^{7} + 31722 p^{2} T^{8} + 2376 p^{3} T^{9} + 246 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 12 T + 18 T^{2} - 286 T^{3} - 1434 T^{4} + 86766 T^{5} - 791061 T^{6} + 86766 p T^{7} - 1434 p^{2} T^{8} - 286 p^{3} T^{9} + 18 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 + 30 T + 513 T^{2} + 5628 T^{3} + 513 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 12 T + 72 T^{2} - 1110 T^{3} + 252 T^{4} + 67596 T^{5} - 176933 T^{6} + 67596 p T^{7} + 252 p^{2} T^{8} - 1110 p^{3} T^{9} + 72 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 255 T^{2} + 38829 T^{4} - 4326059 T^{6} + 38829 p^{2} T^{8} - 255 p^{4} T^{10} + p^{6} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.31508085358425429469870184384, −5.09690202860697081155695974618, −5.00305576220582032563163841512, −4.85770791187052791161851219498, −4.75854093264211091901984835153, −4.61422169397972973560357423153, −4.48879689577358253878960663328, −4.18791283626759020716477967734, −4.04824382670246211824169075526, −4.00475124872872608348487701706, −3.38005933021958014159238031519, −3.30293404103297481809898772647, −3.13707605491251377773733987831, −3.12795788494970790960259596791, −3.05303461850436648812710997491, −2.86946261565772846598474583418, −2.76963380140431549216469073733, −2.70261020987684049333720536684, −2.00547294749686197537005674333, −1.70434621701468172688184840007, −1.62685402800709910634254724062, −1.43875969454305153468034783502, −1.33965864197893331998965466005, −0.45681186331591518945278589535, −0.12442216865366690581766666830, 0.12442216865366690581766666830, 0.45681186331591518945278589535, 1.33965864197893331998965466005, 1.43875969454305153468034783502, 1.62685402800709910634254724062, 1.70434621701468172688184840007, 2.00547294749686197537005674333, 2.70261020987684049333720536684, 2.76963380140431549216469073733, 2.86946261565772846598474583418, 3.05303461850436648812710997491, 3.12795788494970790960259596791, 3.13707605491251377773733987831, 3.30293404103297481809898772647, 3.38005933021958014159238031519, 4.00475124872872608348487701706, 4.04824382670246211824169075526, 4.18791283626759020716477967734, 4.48879689577358253878960663328, 4.61422169397972973560357423153, 4.75854093264211091901984835153, 4.85770791187052791161851219498, 5.00305576220582032563163841512, 5.09690202860697081155695974618, 5.31508085358425429469870184384

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

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