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Properties

Label 12-8330e6-1.1-c1e6-0-3
Degree $12$
Conductor $3.341\times 10^{23}$
Sign $1$
Analytic cond. $8.66029\times 10^{10}$
Root an. cond. $8.15569$
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  − 6·2-s − 2·3-s + 21·4-s + 6·5-s + 12·6-s − 56·8-s − 5·9-s − 36·10-s − 6·11-s − 42·12-s − 12·15-s + 126·16-s + 6·17-s + 30·18-s + 6·19-s + 126·20-s + 36·22-s − 2·23-s + 112·24-s + 21·25-s + 8·27-s − 2·29-s + 72·30-s − 2·31-s − 252·32-s + 12·33-s − 36·34-s + ⋯
L(s)  = 1  − 4.24·2-s − 1.15·3-s + 21/2·4-s + 2.68·5-s + 4.89·6-s − 19.7·8-s − 5/3·9-s − 11.3·10-s − 1.80·11-s − 12.1·12-s − 3.09·15-s + 63/2·16-s + 1.45·17-s + 7.07·18-s + 1.37·19-s + 28.1·20-s + 7.67·22-s − 0.417·23-s + 22.8·24-s + 21/5·25-s + 1.53·27-s − 0.371·29-s + 13.1·30-s − 0.359·31-s − 44.5·32-s + 2.08·33-s − 6.17·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 7^{12} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 7^{12} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 5^{6} \cdot 7^{12} \cdot 17^{6}\)
Sign: $1$
Analytic conductor: \(8.66029\times 10^{10}\)
Root analytic conductor: \(8.15569\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{6} \cdot 5^{6} \cdot 7^{12} \cdot 17^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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)$
bad2 \( ( 1 + T )^{6} \)
5 \( ( 1 - T )^{6} \)
7 \( 1 \)
17 \( ( 1 - T )^{6} \)
good3 \( 1 + 2 T + p^{2} T^{2} + 20 T^{3} + 52 T^{4} + 10 p^{2} T^{5} + 196 T^{6} + 10 p^{3} T^{7} + 52 p^{2} T^{8} + 20 p^{3} T^{9} + p^{6} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 6 T + 73 T^{2} + 320 T^{3} + 192 p T^{4} + 6934 T^{5} + 31462 T^{6} + 6934 p T^{7} + 192 p^{3} T^{8} + 320 p^{3} T^{9} + 73 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 56 T^{2} - 16 T^{3} + 1491 T^{4} - 464 T^{5} + 24296 T^{6} - 464 p T^{7} + 1491 p^{2} T^{8} - 16 p^{3} T^{9} + 56 p^{4} T^{10} + p^{6} T^{12} \)
19 \( 1 - 6 T + 105 T^{2} - 488 T^{3} + 4600 T^{4} - 16918 T^{5} + 112704 T^{6} - 16918 p T^{7} + 4600 p^{2} T^{8} - 488 p^{3} T^{9} + 105 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 2 T + 91 T^{2} + 172 T^{3} + 4218 T^{4} + 6994 T^{5} + 120300 T^{6} + 6994 p T^{7} + 4218 p^{2} T^{8} + 172 p^{3} T^{9} + 91 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 2 T + 63 T^{2} + 68 T^{3} + 532 T^{4} - 2558 T^{5} - 26340 T^{6} - 2558 p T^{7} + 532 p^{2} T^{8} + 68 p^{3} T^{9} + 63 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 2 T + 119 T^{2} + 96 T^{3} + 6992 T^{4} + 3682 T^{5} + 268972 T^{6} + 3682 p T^{7} + 6992 p^{2} T^{8} + 96 p^{3} T^{9} + 119 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 12 T + 4 p T^{2} + 1180 T^{3} + 10715 T^{4} + 70136 T^{5} + 492584 T^{6} + 70136 p T^{7} + 10715 p^{2} T^{8} + 1180 p^{3} T^{9} + 4 p^{5} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 6 T + 175 T^{2} - 748 T^{3} + 13286 T^{4} - 42946 T^{5} + 639452 T^{6} - 42946 p T^{7} + 13286 p^{2} T^{8} - 748 p^{3} T^{9} + 175 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 26 T + 337 T^{2} + 2728 T^{3} + 14414 T^{4} + 43050 T^{5} + 105060 T^{6} + 43050 p T^{7} + 14414 p^{2} T^{8} + 2728 p^{3} T^{9} + 337 p^{4} T^{10} + 26 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 30 T + 561 T^{2} + 7664 T^{3} + 1762 p T^{4} + 736354 T^{5} + 5500114 T^{6} + 736354 p T^{7} + 1762 p^{3} T^{8} + 7664 p^{3} T^{9} + 561 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 14 T + 263 T^{2} - 2424 T^{3} + 29040 T^{4} - 216630 T^{5} + 1972096 T^{6} - 216630 p T^{7} + 29040 p^{2} T^{8} - 2424 p^{3} T^{9} + 263 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 6 T + 217 T^{2} + 1024 T^{3} + 25280 T^{4} + 99398 T^{5} + 1831944 T^{6} + 99398 p T^{7} + 25280 p^{2} T^{8} + 1024 p^{3} T^{9} + 217 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 12 T + 182 T^{2} - 1452 T^{3} + 16347 T^{4} - 146536 T^{5} + 1339884 T^{6} - 146536 p T^{7} + 16347 p^{2} T^{8} - 1452 p^{3} T^{9} + 182 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 14 T + 211 T^{2} + 1012 T^{3} + 10034 T^{4} + 8814 T^{5} + 490572 T^{6} + 8814 p T^{7} + 10034 p^{2} T^{8} + 1012 p^{3} T^{9} + 211 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 12 T + 438 T^{2} - 4132 T^{3} + 78751 T^{4} - 577592 T^{5} + 7482604 T^{6} - 577592 p T^{7} + 78751 p^{2} T^{8} - 4132 p^{3} T^{9} + 438 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 12 T + 404 T^{2} + 3932 T^{3} + 69107 T^{4} + 540952 T^{5} + 6561888 T^{6} + 540952 p T^{7} + 69107 p^{2} T^{8} + 3932 p^{3} T^{9} + 404 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 36 T + 830 T^{2} + 14076 T^{3} + 192959 T^{4} + 2192664 T^{5} + 21106428 T^{6} + 2192664 p T^{7} + 192959 p^{2} T^{8} + 14076 p^{3} T^{9} + 830 p^{4} T^{10} + 36 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 32 T + 538 T^{2} + 5280 T^{3} + 27799 T^{4} - 12192 T^{5} - 1229020 T^{6} - 12192 p T^{7} + 27799 p^{2} T^{8} + 5280 p^{3} T^{9} + 538 p^{4} T^{10} + 32 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 8 T + 324 T^{2} - 2872 T^{3} + 54807 T^{4} - 456256 T^{5} + 5988872 T^{6} - 456256 p T^{7} + 54807 p^{2} T^{8} - 2872 p^{3} T^{9} + 324 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 16 T + 390 T^{2} - 5680 T^{3} + 82367 T^{4} - 917056 T^{5} + 10318580 T^{6} - 917056 p T^{7} + 82367 p^{2} T^{8} - 5680 p^{3} T^{9} + 390 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.55591557545514413419246114445, −4.05679377403588843282982672726, −3.98616654532815545578965250540, −3.73897999454183058137246850147, −3.64712185217725717214951572702, −3.56385218605951935374652307271, −3.41353926002031021213049708489, −3.10508989735173007851258909884, −3.02580721736683241925387772365, −2.98611573378902291408123564700, −2.86136712590927169295518442565, −2.84763076072286798829675971703, −2.76407837152371404163144306246, −2.30480986772603138449921175489, −2.29880086908792558239849059431, −2.09257399927475615525925565819, −2.08880567960249192541715959406, −1.97601301918037862254825906977, −1.70961085973019630005781863749, −1.53055339082637766566522041145, −1.32617253732838521170673444253, −1.23362615717387439353317056312, −1.14274936800527076918652435876, −1.08277486592276530964978367383, −0.953633509304661555685465963063, 0, 0, 0, 0, 0, 0, 0.953633509304661555685465963063, 1.08277486592276530964978367383, 1.14274936800527076918652435876, 1.23362615717387439353317056312, 1.32617253732838521170673444253, 1.53055339082637766566522041145, 1.70961085973019630005781863749, 1.97601301918037862254825906977, 2.08880567960249192541715959406, 2.09257399927475615525925565819, 2.29880086908792558239849059431, 2.30480986772603138449921175489, 2.76407837152371404163144306246, 2.84763076072286798829675971703, 2.86136712590927169295518442565, 2.98611573378902291408123564700, 3.02580721736683241925387772365, 3.10508989735173007851258909884, 3.41353926002031021213049708489, 3.56385218605951935374652307271, 3.64712185217725717214951572702, 3.73897999454183058137246850147, 3.98616654532815545578965250540, 4.05679377403588843282982672726, 4.55591557545514413419246114445

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

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