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Properties

Label 12-85e12-1.1-c1e6-0-9
Degree $12$
Conductor $1.422\times 10^{23}$
Sign $1$
Analytic cond. $3.68714\times 10^{10}$
Root an. cond. $7.59551$
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  − 2·2-s + 4·3-s − 4-s − 8·6-s + 8·7-s + 4·8-s − 4·12-s − 16·14-s − 2·16-s − 12·19-s + 32·21-s + 16·24-s − 28·27-s − 8·28-s − 8·29-s − 4·31-s + 4·32-s + 16·37-s + 24·38-s − 12·41-s − 64·42-s − 8·43-s − 24·47-s − 8·48-s + 6·49-s + 56·54-s + 32·56-s + ⋯
L(s)  = 1  − 1.41·2-s + 2.30·3-s − 1/2·4-s − 3.26·6-s + 3.02·7-s + 1.41·8-s − 1.15·12-s − 4.27·14-s − 1/2·16-s − 2.75·19-s + 6.98·21-s + 3.26·24-s − 5.38·27-s − 1.51·28-s − 1.48·29-s − 0.718·31-s + 0.707·32-s + 2.63·37-s + 3.89·38-s − 1.87·41-s − 9.87·42-s − 1.21·43-s − 3.50·47-s − 1.15·48-s + 6/7·49-s + 7.62·54-s + 4.27·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(5^{12} \cdot 17^{12}\)
Sign: $1$
Analytic conductor: \(3.68714\times 10^{10}\)
Root analytic conductor: \(7.59551\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 5^{12} \cdot 17^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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)$
bad5 \( 1 \)
17 \( 1 \)
good2 \( 1 + p T + 5 T^{2} + p^{3} T^{3} + 15 T^{4} + 9 p T^{5} + 29 T^{6} + 9 p^{2} T^{7} + 15 p^{2} T^{8} + p^{6} T^{9} + 5 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
3 \( 1 - 4 T + 16 T^{2} - 4 p^{2} T^{3} + 85 T^{4} - 140 T^{5} + 278 T^{6} - 140 p T^{7} + 85 p^{2} T^{8} - 4 p^{5} T^{9} + 16 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 8 T + 58 T^{2} - 268 T^{3} + 1117 T^{4} - 3608 T^{5} + 1518 p T^{6} - 3608 p T^{7} + 1117 p^{2} T^{8} - 268 p^{3} T^{9} + 58 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 40 T^{2} + 4 T^{3} + 875 T^{4} + 84 T^{5} + 11786 T^{6} + 84 p T^{7} + 875 p^{2} T^{8} + 4 p^{3} T^{9} + 40 p^{4} T^{10} + p^{6} T^{12} \)
13 \( 1 + 20 T^{2} + 12 T^{3} + 255 T^{4} - 124 T^{5} + 3948 T^{6} - 124 p T^{7} + 255 p^{2} T^{8} + 12 p^{3} T^{9} + 20 p^{4} T^{10} + p^{6} T^{12} \)
19 \( 1 + 12 T + 118 T^{2} + 692 T^{3} + 3655 T^{4} + 14520 T^{5} + 65940 T^{6} + 14520 p T^{7} + 3655 p^{2} T^{8} + 692 p^{3} T^{9} + 118 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 96 T^{2} - 48 T^{3} + 4273 T^{4} - 3380 T^{5} + 119310 T^{6} - 3380 p T^{7} + 4273 p^{2} T^{8} - 48 p^{3} T^{9} + 96 p^{4} T^{10} + p^{6} T^{12} \)
29 \( 1 + 8 T + 120 T^{2} + 824 T^{3} + 7307 T^{4} + 40224 T^{5} + 268432 T^{6} + 40224 p T^{7} + 7307 p^{2} T^{8} + 824 p^{3} T^{9} + 120 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 4 T + 66 T^{2} - 64 T^{3} + 2307 T^{4} - 500 T^{5} + 116326 T^{6} - 500 p T^{7} + 2307 p^{2} T^{8} - 64 p^{3} T^{9} + 66 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 16 T + 232 T^{2} - 2360 T^{3} + 21091 T^{4} - 155176 T^{5} + 1028736 T^{6} - 155176 p T^{7} + 21091 p^{2} T^{8} - 2360 p^{3} T^{9} + 232 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 12 T + 202 T^{2} + 1756 T^{3} + 18495 T^{4} + 123896 T^{5} + 961420 T^{6} + 123896 p T^{7} + 18495 p^{2} T^{8} + 1756 p^{3} T^{9} + 202 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 8 T + 124 T^{2} + 436 T^{3} + 5195 T^{4} - 1788 T^{5} + 132228 T^{6} - 1788 p T^{7} + 5195 p^{2} T^{8} + 436 p^{3} T^{9} + 124 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 24 T + 392 T^{2} + 4556 T^{3} + 43823 T^{4} + 353156 T^{5} + 2577076 T^{6} + 353156 p T^{7} + 43823 p^{2} T^{8} + 4556 p^{3} T^{9} + 392 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 102 T^{2} + 360 T^{3} + 8079 T^{4} + 33592 T^{5} + 448980 T^{6} + 33592 p T^{7} + 8079 p^{2} T^{8} + 360 p^{3} T^{9} + 102 p^{4} T^{10} + p^{6} T^{12} \)
59 \( 1 + 16 T + 362 T^{2} + 4208 T^{3} + 53207 T^{4} + 467872 T^{5} + 4169836 T^{6} + 467872 p T^{7} + 53207 p^{2} T^{8} + 4208 p^{3} T^{9} + 362 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 4 T + 282 T^{2} - 1012 T^{3} + 37143 T^{4} - 112120 T^{5} + 2885772 T^{6} - 112120 p T^{7} + 37143 p^{2} T^{8} - 1012 p^{3} T^{9} + 282 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 4 T + 356 T^{2} - 1240 T^{3} + 55175 T^{4} - 159748 T^{5} + 4798716 T^{6} - 159748 p T^{7} + 55175 p^{2} T^{8} - 1240 p^{3} T^{9} + 356 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 32 T + 620 T^{2} + 8148 T^{3} + 86175 T^{4} + 763340 T^{5} + 6531146 T^{6} + 763340 p T^{7} + 86175 p^{2} T^{8} + 8148 p^{3} T^{9} + 620 p^{4} T^{10} + 32 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 8 T + 262 T^{2} + 920 T^{3} + 23775 T^{4} - 15456 T^{5} + 1456052 T^{6} - 15456 p T^{7} + 23775 p^{2} T^{8} + 920 p^{3} T^{9} + 262 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 36 T + 910 T^{2} + 16320 T^{3} + 234303 T^{4} + 2740860 T^{5} + 26643430 T^{6} + 2740860 p T^{7} + 234303 p^{2} T^{8} + 16320 p^{3} T^{9} + 910 p^{4} T^{10} + 36 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 20 T + 588 T^{2} + 7984 T^{3} + 131023 T^{4} + 1299868 T^{5} + 14803132 T^{6} + 1299868 p T^{7} + 131023 p^{2} T^{8} + 7984 p^{3} T^{9} + 588 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 12 T + 354 T^{2} + 4000 T^{3} + 61415 T^{4} + 604868 T^{5} + 6702448 T^{6} + 604868 p T^{7} + 61415 p^{2} T^{8} + 4000 p^{3} T^{9} + 354 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 16 T + 432 T^{2} - 4376 T^{3} + 78043 T^{4} - 647336 T^{5} + 9312848 T^{6} - 647336 p T^{7} + 78043 p^{2} T^{8} - 4376 p^{3} T^{9} + 432 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.47020475882561516706863001503, −4.45143518268792539690740700465, −4.15144232544469090728635572343, −3.96139866029265066886043231448, −3.84661089557056271562084155446, −3.75566015849714826900067001930, −3.56014020999865573049481077928, −3.46793600608614257647097911906, −3.30306313503399752369565510898, −3.10134885440733272786054552077, −3.08586079020645569281207703041, −2.83146333686028991432911275070, −2.70630871177645902841065793612, −2.70084419405258041578227913713, −2.43953792000200586841426749698, −2.39574118973642221436493037382, −2.17889517564861177590175512224, −1.99571920232625409895755951846, −1.75730910533575659270187283495, −1.67772517744086242557894244120, −1.62407899669038235080827426355, −1.37721985532462679846859204206, −1.32309402360652753363243493078, −1.25569279606045876075109478310, −0.978080463272397225908303454543, 0, 0, 0, 0, 0, 0, 0.978080463272397225908303454543, 1.25569279606045876075109478310, 1.32309402360652753363243493078, 1.37721985532462679846859204206, 1.62407899669038235080827426355, 1.67772517744086242557894244120, 1.75730910533575659270187283495, 1.99571920232625409895755951846, 2.17889517564861177590175512224, 2.39574118973642221436493037382, 2.43953792000200586841426749698, 2.70084419405258041578227913713, 2.70630871177645902841065793612, 2.83146333686028991432911275070, 3.08586079020645569281207703041, 3.10134885440733272786054552077, 3.30306313503399752369565510898, 3.46793600608614257647097911906, 3.56014020999865573049481077928, 3.75566015849714826900067001930, 3.84661089557056271562084155446, 3.96139866029265066886043231448, 4.15144232544469090728635572343, 4.45143518268792539690740700465, 4.47020475882561516706863001503

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

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