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Properties

Label 12-4018e6-1.1-c1e6-0-2
Degree $12$
Conductor $4.208\times 10^{21}$
Sign $1$
Analytic cond. $1.09074\times 10^{9}$
Root an. cond. $5.66426$
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  + 6·2-s + 4·3-s + 21·4-s + 12·5-s + 24·6-s + 56·8-s + 8·9-s + 72·10-s + 84·12-s + 8·13-s + 48·15-s + 126·16-s + 16·17-s + 48·18-s + 12·19-s + 252·20-s + 12·23-s + 224·24-s + 64·25-s + 48·26-s + 16·27-s + 288·30-s − 4·31-s + 252·32-s + 96·34-s + 168·36-s − 24·37-s + ⋯
L(s)  = 1  + 4.24·2-s + 2.30·3-s + 21/2·4-s + 5.36·5-s + 9.79·6-s + 19.7·8-s + 8/3·9-s + 22.7·10-s + 24.2·12-s + 2.21·13-s + 12.3·15-s + 63/2·16-s + 3.88·17-s + 11.3·18-s + 2.75·19-s + 56.3·20-s + 2.50·23-s + 45.7·24-s + 64/5·25-s + 9.41·26-s + 3.07·27-s + 52.5·30-s − 0.718·31-s + 44.5·32-s + 16.4·34-s + 28·36-s − 3.94·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{12} \cdot 41^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{12} \cdot 41^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 7^{12} \cdot 41^{6}\)
Sign: $1$
Analytic conductor: \(1.09074\times 10^{9}\)
Root analytic conductor: \(5.66426\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 7^{12} \cdot 41^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(7675.352097\)
\(L(\frac12)\) \(\approx\) \(7675.352097\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T )^{6} \)
7 \( 1 \)
41 \( ( 1 + T )^{6} \)
good3 \( 1 - 4 T + 8 T^{2} - 16 T^{3} + 35 T^{4} - 68 T^{5} + 124 T^{6} - 68 p T^{7} + 35 p^{2} T^{8} - 16 p^{3} T^{9} + 8 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 - 12 T + 16 p T^{2} - 368 T^{3} + 1307 T^{4} - 3772 T^{5} + 9162 T^{6} - 3772 p T^{7} + 1307 p^{2} T^{8} - 368 p^{3} T^{9} + 16 p^{5} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 16 T^{2} + 8 T^{3} + 113 T^{4} - 36 T^{5} + 878 T^{6} - 36 p T^{7} + 113 p^{2} T^{8} + 8 p^{3} T^{9} + 16 p^{4} T^{10} + p^{6} T^{12} \)
13 \( 1 - 8 T + 72 T^{2} - 404 T^{3} + 175 p T^{4} - 9484 T^{5} + 38732 T^{6} - 9484 p T^{7} + 175 p^{3} T^{8} - 404 p^{3} T^{9} + 72 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 16 T + 156 T^{2} - 1092 T^{3} + 6311 T^{4} - 31084 T^{5} + 136444 T^{6} - 31084 p T^{7} + 6311 p^{2} T^{8} - 1092 p^{3} T^{9} + 156 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 12 T + 128 T^{2} - 952 T^{3} + 6203 T^{4} - 32708 T^{5} + 157212 T^{6} - 32708 p T^{7} + 6203 p^{2} T^{8} - 952 p^{3} T^{9} + 128 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 12 T + 134 T^{2} - 992 T^{3} + 7183 T^{4} - 40036 T^{5} + 214808 T^{6} - 40036 p T^{7} + 7183 p^{2} T^{8} - 992 p^{3} T^{9} + 134 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 96 T^{2} + 48 T^{3} + 4853 T^{4} + 3300 T^{5} + 165406 T^{6} + 3300 p T^{7} + 4853 p^{2} T^{8} + 48 p^{3} T^{9} + 96 p^{4} T^{10} + p^{6} T^{12} \)
31 \( 1 + 4 T + 96 T^{2} + 492 T^{3} + 5139 T^{4} + 23896 T^{5} + 194656 T^{6} + 23896 p T^{7} + 5139 p^{2} T^{8} + 492 p^{3} T^{9} + 96 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 24 T + 398 T^{2} + 4600 T^{3} + 43895 T^{4} + 339152 T^{5} + 2260068 T^{6} + 339152 p T^{7} + 43895 p^{2} T^{8} + 4600 p^{3} T^{9} + 398 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 4 T + 150 T^{2} - 708 T^{3} + 11959 T^{4} - 55888 T^{5} + 614084 T^{6} - 55888 p T^{7} + 11959 p^{2} T^{8} - 708 p^{3} T^{9} + 150 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 16 T + 256 T^{2} + 2652 T^{3} + 27239 T^{4} + 215852 T^{5} + 1650068 T^{6} + 215852 p T^{7} + 27239 p^{2} T^{8} + 2652 p^{3} T^{9} + 256 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 16 T + 234 T^{2} + 1596 T^{3} + 10025 T^{4} + 6328 T^{5} + 37970 T^{6} + 6328 p T^{7} + 10025 p^{2} T^{8} + 1596 p^{3} T^{9} + 234 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 16 T + 120 T^{2} - 1140 T^{3} + 13555 T^{4} - 89804 T^{5} + 500490 T^{6} - 89804 p T^{7} + 13555 p^{2} T^{8} - 1140 p^{3} T^{9} + 120 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 32 T + 726 T^{2} - 11300 T^{3} + 143695 T^{4} - 1456668 T^{5} + 12552758 T^{6} - 1456668 p T^{7} + 143695 p^{2} T^{8} - 11300 p^{3} T^{9} + 726 p^{4} T^{10} - 32 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 20 T + 362 T^{2} + 4304 T^{3} + 50513 T^{4} + 485936 T^{5} + 4399546 T^{6} + 485936 p T^{7} + 50513 p^{2} T^{8} + 4304 p^{3} T^{9} + 362 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 12 T + 334 T^{2} - 2596 T^{3} + 45983 T^{4} - 276920 T^{5} + 3969444 T^{6} - 276920 p T^{7} + 45983 p^{2} T^{8} - 2596 p^{3} T^{9} + 334 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 16 T + 512 T^{2} - 5856 T^{3} + 101027 T^{4} - 855728 T^{5} + 10072096 T^{6} - 855728 p T^{7} + 101027 p^{2} T^{8} - 5856 p^{3} T^{9} + 512 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 386 T^{2} - 288 T^{3} + 65775 T^{4} - 67872 T^{5} + 6560348 T^{6} - 67872 p T^{7} + 65775 p^{2} T^{8} - 288 p^{3} T^{9} + 386 p^{4} T^{10} + p^{6} T^{12} \)
83 \( 1 - 32 T + 632 T^{2} - 9580 T^{3} + 121787 T^{4} - 1345308 T^{5} + 13167754 T^{6} - 1345308 p T^{7} + 121787 p^{2} T^{8} - 9580 p^{3} T^{9} + 632 p^{4} T^{10} - 32 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 8 T + 406 T^{2} - 3056 T^{3} + 76583 T^{4} - 508264 T^{5} + 8599604 T^{6} - 508264 p T^{7} + 76583 p^{2} T^{8} - 3056 p^{3} T^{9} + 406 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 8 T + 142 T^{2} - 584 T^{3} + 14639 T^{4} - 14000 T^{5} + 886340 T^{6} - 14000 p T^{7} + 14639 p^{2} T^{8} - 584 p^{3} T^{9} + 142 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.32443162945244366113461941276, −4.10922814282366197653615335485, −3.74191863720047945034800232587, −3.72831263083728642040709692620, −3.69451138722700345070662655043, −3.61791283737427072335146662218, −3.46931329183545669341853766231, −3.35701742407016042473028126172, −3.23889444685784016296265409609, −3.00686443897721175969339370601, −2.95604806192576808530986240153, −2.89090984441971631533693659201, −2.86119593800416600264567618366, −2.28450344428187024437901068213, −2.24749958297416509134274181232, −2.21121256554818720122840266651, −2.17727802723836070404764206846, −1.97279574674611994022158667882, −1.65703740192490960787302199229, −1.64687067073216532921882316046, −1.29563164919179036971628736294, −1.26020124105450549697162550433, −1.17598103046343536802250042185, −1.08425862357158940386181237072, −0.867467566043298093277351910010, 0.867467566043298093277351910010, 1.08425862357158940386181237072, 1.17598103046343536802250042185, 1.26020124105450549697162550433, 1.29563164919179036971628736294, 1.64687067073216532921882316046, 1.65703740192490960787302199229, 1.97279574674611994022158667882, 2.17727802723836070404764206846, 2.21121256554818720122840266651, 2.24749958297416509134274181232, 2.28450344428187024437901068213, 2.86119593800416600264567618366, 2.89090984441971631533693659201, 2.95604806192576808530986240153, 3.00686443897721175969339370601, 3.23889444685784016296265409609, 3.35701742407016042473028126172, 3.46931329183545669341853766231, 3.61791283737427072335146662218, 3.69451138722700345070662655043, 3.72831263083728642040709692620, 3.74191863720047945034800232587, 4.10922814282366197653615335485, 4.32443162945244366113461941276

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

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