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Properties

Label 12-21e12-1.1-c1e6-0-9
Degree $12$
Conductor $7.356\times 10^{15}$
Sign $1$
Analytic cond. $1906.75$
Root an. cond. $1.87654$
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 − 6·5-s + 7·7-s − 18·10-s − 2·11-s + 21·14-s − 4·17-s − 8·19-s − 12·20-s − 6·22-s + 10·23-s + 26·25-s + 14·28-s + 16·29-s − 10·31-s − 14·32-s − 12·34-s − 42·35-s + 4·37-s − 24·38-s + 12·43-s − 4·44-s + 30·46-s + 15·47-s + 28·49-s + 78·50-s + ⋯
L(s)  = 1  + 2.12·2-s + 4-s − 2.68·5-s + 2.64·7-s − 5.69·10-s − 0.603·11-s + 5.61·14-s − 0.970·17-s − 1.83·19-s − 2.68·20-s − 1.27·22-s + 2.08·23-s + 26/5·25-s + 2.64·28-s + 2.97·29-s − 1.79·31-s − 2.47·32-s − 2.05·34-s − 7.09·35-s + 0.657·37-s − 3.89·38-s + 1.82·43-s − 0.603·44-s + 4.42·46-s + 2.18·47-s + 4·49-s + 11.0·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(1906.75\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(7.661910741\)
\(L(\frac12)\) \(\approx\) \(7.661910741\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 - p T + 3 p T^{2} - p^{2} T^{3} + 3 p^{2} T^{4} - p^{3} T^{5} + p^{3} T^{6} \)
good2 \( 1 - 3 T + 7 T^{2} - 15 T^{3} + 31 T^{4} - 49 T^{5} + 71 T^{6} - 49 p T^{7} + 31 p^{2} T^{8} - 15 p^{3} T^{9} + 7 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 + 6 T + 2 p T^{2} - p T^{3} - 3 T^{4} + 161 T^{5} + 561 T^{6} + 161 p T^{7} - 3 p^{2} T^{8} - p^{4} T^{9} + 2 p^{5} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 2 T - 14 T^{2} - 9 p T^{3} - 177 T^{4} + 707 T^{5} + 5559 T^{6} + 707 p T^{7} - 177 p^{2} T^{8} - 9 p^{4} T^{9} - 14 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + T^{2} - 14 T^{3} + 43 T^{4} + 602 T^{5} - 209 T^{6} + 602 p T^{7} + 43 p^{2} T^{8} - 14 p^{3} T^{9} + p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 + 4 T - 15 T^{2} - 226 T^{3} - 537 T^{4} + 2198 T^{5} + 21911 T^{6} + 2198 p T^{7} - 537 p^{2} T^{8} - 226 p^{3} T^{9} - 15 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( ( 1 + 4 T + 46 T^{2} + 151 T^{3} + 46 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 10 T + 7 T^{2} + 300 T^{3} - 1131 T^{4} - 5110 T^{5} + 53943 T^{6} - 5110 p T^{7} - 1131 p^{2} T^{8} + 300 p^{3} T^{9} + 7 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 16 T + 129 T^{2} - 914 T^{3} + 5787 T^{4} - 1042 p T^{5} + 154847 T^{6} - 1042 p^{2} T^{7} + 5787 p^{2} T^{8} - 914 p^{3} T^{9} + 129 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 + 5 T + 99 T^{2} + 311 T^{3} + 99 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 4 T + 35 T^{2} + 134 T^{3} + 1193 T^{4} - 3934 T^{5} + 120891 T^{6} - 3934 p T^{7} + 1193 p^{2} T^{8} + 134 p^{3} T^{9} + 35 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 15 T^{2} - 112 T^{3} + 1317 T^{4} - 3178 T^{5} - 18717 T^{6} - 3178 p T^{7} + 1317 p^{2} T^{8} - 112 p^{3} T^{9} + 15 p^{4} T^{10} + p^{6} T^{12} \)
43 \( 1 - 12 T + 31 T^{2} - 136 T^{3} + 2539 T^{4} - 5930 T^{5} - 45857 T^{6} - 5930 p T^{7} + 2539 p^{2} T^{8} - 136 p^{3} T^{9} + 31 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 15 T + 115 T^{2} - 985 T^{3} + 10854 T^{4} - 81956 T^{5} + 533835 T^{6} - 81956 p T^{7} + 10854 p^{2} T^{8} - 985 p^{3} T^{9} + 115 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 26 T + 343 T^{2} - 2892 T^{3} + 17925 T^{4} - 100058 T^{5} + 632955 T^{6} - 100058 p T^{7} + 17925 p^{2} T^{8} - 2892 p^{3} T^{9} + 343 p^{4} T^{10} - 26 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 11 T + 6 T^{2} - 10 p T^{3} - 2637 T^{4} + 28819 T^{5} + 423452 T^{6} + 28819 p T^{7} - 2637 p^{2} T^{8} - 10 p^{4} T^{9} + 6 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 8 T - 18 T^{2} - 359 T^{3} + 1229 T^{4} - 16569 T^{5} - 310715 T^{6} - 16569 p T^{7} + 1229 p^{2} T^{8} - 359 p^{3} T^{9} - 18 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
67 \( ( 1 + 6 T + 206 T^{2} + 791 T^{3} + 206 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
71 \( 1 + 5 T + 66 T^{2} - 116 T^{3} + 4296 T^{4} - 31079 T^{5} + 6755 T^{6} - 31079 p T^{7} + 4296 p^{2} T^{8} - 116 p^{3} T^{9} + 66 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 4 T + 111 T^{2} - 68 T^{3} + 11531 T^{4} + 24234 T^{5} + 541591 T^{6} + 24234 p T^{7} + 11531 p^{2} T^{8} - 68 p^{3} T^{9} + 111 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
79 \( ( 1 - 30 T + 530 T^{2} - 5677 T^{3} + 530 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 14 T + 169 T^{2} - 2604 T^{3} + 26195 T^{4} - 277228 T^{5} + 3109373 T^{6} - 277228 p T^{7} + 26195 p^{2} T^{8} - 2604 p^{3} T^{9} + 169 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 13 T + 192 T^{2} + 2018 T^{3} + 26268 T^{4} + 3031 p T^{5} + 3321179 T^{6} + 3031 p^{2} T^{7} + 26268 p^{2} T^{8} + 2018 p^{3} T^{9} + 192 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 + 284 T^{2} - 7 T^{3} + 284 p T^{4} + p^{3} T^{6} )^{2} \)
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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

−5.85940698467189243318979863339, −5.63822204577503443275232497041, −5.56298069381681777774212712291, −5.21702598209138893820224473424, −5.18513358002557348896410734431, −4.73839307659248023967468252687, −4.69080974514985826374556831385, −4.65407160808077856731974626160, −4.50770393053126374422946010113, −4.47720353053565604083886561068, −4.46522111264836069283533552594, −4.37000963961824166690264547728, −3.69689550044341404531735979624, −3.63100818207655969816269306957, −3.60376678114495628757972271468, −3.38889224175851233801621439773, −3.23948318219134329682124854637, −2.60363442661833080602422101610, −2.37783180492650320097812825296, −2.21956291785853385926931319298, −2.21765084941969900090748997822, −1.83115924420445811066180626022, −0.969023841601503680259129312344, −0.860673458010605239564689985345, −0.70661006873585351337156361840, 0.70661006873585351337156361840, 0.860673458010605239564689985345, 0.969023841601503680259129312344, 1.83115924420445811066180626022, 2.21765084941969900090748997822, 2.21956291785853385926931319298, 2.37783180492650320097812825296, 2.60363442661833080602422101610, 3.23948318219134329682124854637, 3.38889224175851233801621439773, 3.60376678114495628757972271468, 3.63100818207655969816269306957, 3.69689550044341404531735979624, 4.37000963961824166690264547728, 4.46522111264836069283533552594, 4.47720353053565604083886561068, 4.50770393053126374422946010113, 4.65407160808077856731974626160, 4.69080974514985826374556831385, 4.73839307659248023967468252687, 5.18513358002557348896410734431, 5.21702598209138893820224473424, 5.56298069381681777774212712291, 5.63822204577503443275232497041, 5.85940698467189243318979863339

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

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