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Properties

Label 12-2415e6-1.1-c1e6-0-1
Degree $12$
Conductor $1.984\times 10^{20}$
Sign $1$
Analytic cond. $5.14239\times 10^{7}$
Root an. cond. $4.39134$
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 − 6·3-s + 4-s − 6·5-s − 18·6-s + 6·7-s − 5·8-s + 21·9-s − 18·10-s + 6·11-s − 6·12-s − 6·13-s + 18·14-s + 36·15-s − 3·16-s + 8·17-s + 63·18-s − 8·19-s − 6·20-s − 36·21-s + 18·22-s − 6·23-s + 30·24-s + 21·25-s − 18·26-s − 56·27-s + 6·28-s + ⋯
L(s)  = 1  + 2.12·2-s − 3.46·3-s + 1/2·4-s − 2.68·5-s − 7.34·6-s + 2.26·7-s − 1.76·8-s + 7·9-s − 5.69·10-s + 1.80·11-s − 1.73·12-s − 1.66·13-s + 4.81·14-s + 9.29·15-s − 3/4·16-s + 1.94·17-s + 14.8·18-s − 1.83·19-s − 1.34·20-s − 7.85·21-s + 3.83·22-s − 1.25·23-s + 6.12·24-s + 21/5·25-s − 3.53·26-s − 10.7·27-s + 1.13·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 7^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 7^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 5^{6} \cdot 7^{6} \cdot 23^{6}\)
Sign: $1$
Analytic conductor: \(5.14239\times 10^{7}\)
Root analytic conductor: \(4.39134\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 5^{6} \cdot 7^{6} \cdot 23^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.432247548\)
\(L(\frac12)\) \(\approx\) \(5.432247548\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T )^{6} \)
5 \( ( 1 + T )^{6} \)
7 \( ( 1 - T )^{6} \)
23 \( ( 1 + T )^{6} \)
good2 \( 1 - 3 T + p^{3} T^{2} - p^{4} T^{3} + 7 p^{2} T^{4} - 45 T^{5} + 67 T^{6} - 45 p T^{7} + 7 p^{4} T^{8} - p^{7} T^{9} + p^{7} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 6 T + 51 T^{2} - 212 T^{3} + 1110 T^{4} - 3726 T^{5} + 15010 T^{6} - 3726 p T^{7} + 1110 p^{2} T^{8} - 212 p^{3} T^{9} + 51 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 6 T + 47 T^{2} + 216 T^{3} + 1106 T^{4} + 4630 T^{5} + 17962 T^{6} + 4630 p T^{7} + 1106 p^{2} T^{8} + 216 p^{3} T^{9} + 47 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 8 T + 91 T^{2} - 516 T^{3} + 3320 T^{4} - 14792 T^{5} + 4136 p T^{6} - 14792 p T^{7} + 3320 p^{2} T^{8} - 516 p^{3} T^{9} + 91 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 8 T + 89 T^{2} + 538 T^{3} + 3706 T^{4} + 17614 T^{5} + 89372 T^{6} + 17614 p T^{7} + 3706 p^{2} T^{8} + 538 p^{3} T^{9} + 89 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 8 T + 134 T^{2} - 912 T^{3} + 8375 T^{4} - 47304 T^{5} + 309780 T^{6} - 47304 p T^{7} + 8375 p^{2} T^{8} - 912 p^{3} T^{9} + 134 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 16 T + 200 T^{2} + 1688 T^{3} + 13543 T^{4} + 88904 T^{5} + 550592 T^{6} + 88904 p T^{7} + 13543 p^{2} T^{8} + 1688 p^{3} T^{9} + 200 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 8 T + 83 T^{2} - 762 T^{3} + 6836 T^{4} - 43070 T^{5} + 279278 T^{6} - 43070 p T^{7} + 6836 p^{2} T^{8} - 762 p^{3} T^{9} + 83 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 8 T + 157 T^{2} - 502 T^{3} + 7402 T^{4} + 3162 T^{5} + 231856 T^{6} + 3162 p T^{7} + 7402 p^{2} T^{8} - 502 p^{3} T^{9} + 157 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 8 T + 141 T^{2} - 630 T^{3} + 7154 T^{4} - 21470 T^{5} + 281032 T^{6} - 21470 p T^{7} + 7154 p^{2} T^{8} - 630 p^{3} T^{9} + 141 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 10 T + 194 T^{2} - 1358 T^{3} + 17311 T^{4} - 98308 T^{5} + 988156 T^{6} - 98308 p T^{7} + 17311 p^{2} T^{8} - 1358 p^{3} T^{9} + 194 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 28 T + 472 T^{2} - 5916 T^{3} + 61391 T^{4} - 537928 T^{5} + 4159168 T^{6} - 537928 p T^{7} + 61391 p^{2} T^{8} - 5916 p^{3} T^{9} + 472 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 6 T + 201 T^{2} + 806 T^{3} + 19488 T^{4} + 64800 T^{5} + 1342514 T^{6} + 64800 p T^{7} + 19488 p^{2} T^{8} + 806 p^{3} T^{9} + 201 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 12 T + 197 T^{2} + 1374 T^{3} + 15506 T^{4} + 80782 T^{5} + 937036 T^{6} + 80782 p T^{7} + 15506 p^{2} T^{8} + 1374 p^{3} T^{9} + 197 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 18 T + 227 T^{2} - 1642 T^{3} + 14822 T^{4} - 132888 T^{5} + 1350740 T^{6} - 132888 p T^{7} + 14822 p^{2} T^{8} - 1642 p^{3} T^{9} + 227 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 10 T + 262 T^{2} - 2310 T^{3} + 36575 T^{4} - 272812 T^{5} + 3183700 T^{6} - 272812 p T^{7} + 36575 p^{2} T^{8} - 2310 p^{3} T^{9} + 262 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 2 T + 335 T^{2} + 812 T^{3} + 51414 T^{4} + 124278 T^{5} + 4711538 T^{6} + 124278 p T^{7} + 51414 p^{2} T^{8} + 812 p^{3} T^{9} + 335 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 2 T + 284 T^{2} - 166 T^{3} + 39111 T^{4} - 64056 T^{5} + 3659672 T^{6} - 64056 p T^{7} + 39111 p^{2} T^{8} - 166 p^{3} T^{9} + 284 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 26 T + 381 T^{2} - 4544 T^{3} + 52300 T^{4} - 578450 T^{5} + 5751952 T^{6} - 578450 p T^{7} + 52300 p^{2} T^{8} - 4544 p^{3} T^{9} + 381 p^{4} T^{10} - 26 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 8 T + 302 T^{2} - 2512 T^{3} + 44719 T^{4} - 398408 T^{5} + 4621732 T^{6} - 398408 p T^{7} + 44719 p^{2} T^{8} - 2512 p^{3} T^{9} + 302 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 20 T + 354 T^{2} - 3268 T^{3} + 39631 T^{4} - 344168 T^{5} + 4353756 T^{6} - 344168 p T^{7} + 39631 p^{2} T^{8} - 3268 p^{3} T^{9} + 354 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.69356661504649976984989661874, −4.37167748121079573954941955894, −4.36440290800721127327589993157, −4.26297397015875186113455821645, −4.20434201442919585257294627343, −4.19224798362072201330711638922, −4.11804113842116677795718915829, −3.65352102229116868393864085066, −3.63063555067820735488778587531, −3.58129308924506522184811909972, −3.39498666756076663476993396330, −3.38917975884962752974233196060, −2.90170949905393631870208395875, −2.47537721623582506708645740084, −2.43269079783713073900556519190, −2.21397479752081841353261513419, −2.13339056573458886881262046587, −1.74124927566829554642632483060, −1.67309778722328275382559028918, −1.32765511466818081337097514675, −1.03445475601234459343794439501, −0.71221873729383407099206595559, −0.66271853547962588883432661710, −0.53286283152280864738514621979, −0.50620076718392544027185222817, 0.50620076718392544027185222817, 0.53286283152280864738514621979, 0.66271853547962588883432661710, 0.71221873729383407099206595559, 1.03445475601234459343794439501, 1.32765511466818081337097514675, 1.67309778722328275382559028918, 1.74124927566829554642632483060, 2.13339056573458886881262046587, 2.21397479752081841353261513419, 2.43269079783713073900556519190, 2.47537721623582506708645740084, 2.90170949905393631870208395875, 3.38917975884962752974233196060, 3.39498666756076663476993396330, 3.58129308924506522184811909972, 3.63063555067820735488778587531, 3.65352102229116868393864085066, 4.11804113842116677795718915829, 4.19224798362072201330711638922, 4.20434201442919585257294627343, 4.26297397015875186113455821645, 4.36440290800721127327589993157, 4.37167748121079573954941955894, 4.69356661504649976984989661874

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

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