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Properties

Label 12-51e12-1.1-c1e6-0-3
Degree $12$
Conductor $3.096\times 10^{20}$
Sign $1$
Analytic cond. $8.02610\times 10^{7}$
Root an. cond. $4.55731$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 3·4-s − 3·5-s − 3·7-s − 2·8-s + 9·10-s − 9·11-s + 9·13-s + 9·14-s + 3·16-s + 9·19-s − 9·20-s + 27·22-s − 9·23-s − 3·25-s − 27·26-s − 9·28-s − 6·29-s − 24·31-s − 6·32-s + 9·35-s + 3·37-s − 27·38-s + 6·40-s − 18·41-s − 27·44-s + 27·46-s + ⋯
L(s)  = 1  − 2.12·2-s + 3/2·4-s − 1.34·5-s − 1.13·7-s − 0.707·8-s + 2.84·10-s − 2.71·11-s + 2.49·13-s + 2.40·14-s + 3/4·16-s + 2.06·19-s − 2.01·20-s + 5.75·22-s − 1.87·23-s − 3/5·25-s − 5.29·26-s − 1.70·28-s − 1.11·29-s − 4.31·31-s − 1.06·32-s + 1.52·35-s + 0.493·37-s − 4.37·38-s + 0.948·40-s − 2.81·41-s − 4.07·44-s + 3.98·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 17^{12}\)
Sign: $1$
Analytic conductor: \(8.02610\times 10^{7}\)
Root analytic conductor: \(4.55731\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 3^{12} \cdot 17^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + 3 T + 3 p T^{2} + 11 T^{3} + 9 p T^{4} + 15 p T^{5} + 3 p^{4} T^{6} + 15 p^{2} T^{7} + 9 p^{3} T^{8} + 11 p^{3} T^{9} + 3 p^{5} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 + 3 T + 12 T^{2} + 24 T^{3} + 3 p^{2} T^{4} + 213 T^{5} + 536 T^{6} + 213 p T^{7} + 3 p^{4} T^{8} + 24 p^{3} T^{9} + 12 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 3 T + 15 T^{2} + 30 T^{3} + 150 T^{4} + 192 T^{5} + 839 T^{6} + 192 p T^{7} + 150 p^{2} T^{8} + 30 p^{3} T^{9} + 15 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 9 T + 60 T^{2} + 296 T^{3} + 1269 T^{4} + 4755 T^{5} + 16356 T^{6} + 4755 p T^{7} + 1269 p^{2} T^{8} + 296 p^{3} T^{9} + 60 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 9 T + 87 T^{2} - 482 T^{3} + 2688 T^{4} - 10872 T^{5} + 44767 T^{6} - 10872 p T^{7} + 2688 p^{2} T^{8} - 482 p^{3} T^{9} + 87 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 9 T + 105 T^{2} - 648 T^{3} + 240 p T^{4} - 21474 T^{5} + 111521 T^{6} - 21474 p T^{7} + 240 p^{3} T^{8} - 648 p^{3} T^{9} + 105 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 9 T + 102 T^{2} + 558 T^{3} + 4053 T^{4} + 17037 T^{5} + 104432 T^{6} + 17037 p T^{7} + 4053 p^{2} T^{8} + 558 p^{3} T^{9} + 102 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 6 T + 171 T^{2} + 813 T^{3} + 12231 T^{4} + 45633 T^{5} + 470690 T^{6} + 45633 p T^{7} + 12231 p^{2} T^{8} + 813 p^{3} T^{9} + 171 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 24 T + 12 p T^{2} + 4226 T^{3} + 37860 T^{4} + 277500 T^{5} + 1691791 T^{6} + 277500 p T^{7} + 37860 p^{2} T^{8} + 4226 p^{3} T^{9} + 12 p^{5} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 3 T + 147 T^{2} - 274 T^{3} + 9672 T^{4} - 10032 T^{5} + 414439 T^{6} - 10032 p T^{7} + 9672 p^{2} T^{8} - 274 p^{3} T^{9} + 147 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 18 T + 309 T^{2} + 3267 T^{3} + 32637 T^{4} + 245871 T^{5} + 1773626 T^{6} + 245871 p T^{7} + 32637 p^{2} T^{8} + 3267 p^{3} T^{9} + 309 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 141 T^{2} + 297 T^{3} + 9951 T^{4} + 33021 T^{5} + 481646 T^{6} + 33021 p T^{7} + 9951 p^{2} T^{8} + 297 p^{3} T^{9} + 141 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 + 24 T + 453 T^{2} + 5671 T^{3} + 61935 T^{4} + 530319 T^{5} + 4048494 T^{6} + 530319 p T^{7} + 61935 p^{2} T^{8} + 5671 p^{3} T^{9} + 453 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 24 T + 489 T^{2} + 6455 T^{3} + 75489 T^{4} + 681369 T^{5} + 5546778 T^{6} + 681369 p T^{7} + 75489 p^{2} T^{8} + 6455 p^{3} T^{9} + 489 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 9 T + 294 T^{2} - 2214 T^{3} + 38787 T^{4} - 239589 T^{5} + 2942228 T^{6} - 239589 p T^{7} + 38787 p^{2} T^{8} - 2214 p^{3} T^{9} + 294 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 21 T + 411 T^{2} + 5420 T^{3} + 63636 T^{4} + 611928 T^{5} + 5190733 T^{6} + 611928 p T^{7} + 63636 p^{2} T^{8} + 5420 p^{3} T^{9} + 411 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 6 T + 249 T^{2} + 1819 T^{3} + 30213 T^{4} + 235809 T^{5} + 2399994 T^{6} + 235809 p T^{7} + 30213 p^{2} T^{8} + 1819 p^{3} T^{9} + 249 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 27 T + 633 T^{2} + 9886 T^{3} + 131955 T^{4} + 1415775 T^{5} + 13062822 T^{6} + 1415775 p T^{7} + 131955 p^{2} T^{8} + 9886 p^{3} T^{9} + 633 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 18 T + 408 T^{2} + 4781 T^{3} + 66651 T^{4} + 601068 T^{5} + 6253045 T^{6} + 601068 p T^{7} + 66651 p^{2} T^{8} + 4781 p^{3} T^{9} + 408 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 24 T + 483 T^{2} + 7512 T^{3} + 96486 T^{4} + 1056696 T^{5} + 10233479 T^{6} + 1056696 p T^{7} + 96486 p^{2} T^{8} + 7512 p^{3} T^{9} + 483 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 6 T + 255 T^{2} + 911 T^{3} + 33621 T^{4} + 66129 T^{5} + 3005934 T^{6} + 66129 p T^{7} + 33621 p^{2} T^{8} + 911 p^{3} T^{9} + 255 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 207 T^{2} - 585 T^{3} + 18153 T^{4} - 157635 T^{5} + 1321982 T^{6} - 157635 p T^{7} + 18153 p^{2} T^{8} - 585 p^{3} T^{9} + 207 p^{4} T^{10} + p^{6} T^{12} \)
97 \( 1 - 33 T + 906 T^{2} - 16414 T^{3} + 259359 T^{4} - 3196683 T^{5} + 35074495 T^{6} - 3196683 p T^{7} + 259359 p^{2} T^{8} - 16414 p^{3} T^{9} + 906 p^{4} T^{10} - 33 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.13381985168945127356757719382, −4.83227634469482461550653045691, −4.60308636331657089248095392796, −4.53362080736538880752693476658, −4.52969244588324898620353501978, −4.45136006218563457605141965814, −4.13297135203975348008925373993, −3.79088041201331679649847508804, −3.69939221874558771610468879578, −3.55660627698293781390751309636, −3.46002484639015166020376970217, −3.32669806841304530423702794380, −3.21513898578496158303837290689, −3.18829525060057467753163926667, −3.06267541541045915163594609253, −3.00252751878126545453657420363, −2.49241697735560912052771671527, −2.23983813006014734914676685648, −2.14499882028126511109749377681, −1.88298424106668734623719557020, −1.71094289357146942503382595001, −1.53542126069137570400270125614, −1.41312202047877082228663778035, −1.27271166643478354718172822741, −0.985254837824371413532214063994, 0, 0, 0, 0, 0, 0, 0.985254837824371413532214063994, 1.27271166643478354718172822741, 1.41312202047877082228663778035, 1.53542126069137570400270125614, 1.71094289357146942503382595001, 1.88298424106668734623719557020, 2.14499882028126511109749377681, 2.23983813006014734914676685648, 2.49241697735560912052771671527, 3.00252751878126545453657420363, 3.06267541541045915163594609253, 3.18829525060057467753163926667, 3.21513898578496158303837290689, 3.32669806841304530423702794380, 3.46002484639015166020376970217, 3.55660627698293781390751309636, 3.69939221874558771610468879578, 3.79088041201331679649847508804, 4.13297135203975348008925373993, 4.45136006218563457605141965814, 4.52969244588324898620353501978, 4.53362080736538880752693476658, 4.60308636331657089248095392796, 4.83227634469482461550653045691, 5.13381985168945127356757719382

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

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