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Properties

Label 12-1323e6-1.1-c1e6-0-0
Degree $12$
Conductor $5.362\times 10^{18}$
Sign $1$
Analytic cond. $1.39002\times 10^{6}$
Root an. cond. $3.25026$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 6·4-s + 6·5-s + 9·8-s + 18·10-s − 12·11-s − 3·13-s + 12·16-s − 6·17-s − 3·19-s + 36·20-s − 36·22-s − 24·23-s − 3·25-s − 9·26-s + 9·29-s − 3·31-s + 12·32-s − 18·34-s + 3·37-s − 9·38-s + 54·40-s + 3·43-s − 72·44-s − 72·46-s − 3·47-s − 9·50-s + ⋯
L(s)  = 1  + 2.12·2-s + 3·4-s + 2.68·5-s + 3.18·8-s + 5.69·10-s − 3.61·11-s − 0.832·13-s + 3·16-s − 1.45·17-s − 0.688·19-s + 8.04·20-s − 7.67·22-s − 5.00·23-s − 3/5·25-s − 1.76·26-s + 1.67·29-s − 0.538·31-s + 2.12·32-s − 3.08·34-s + 0.493·37-s − 1.45·38-s + 8.53·40-s + 0.457·43-s − 10.8·44-s − 10.6·46-s − 0.437·47-s − 1.27·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{18} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(1.39002\times 10^{6}\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{18} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1454051105\)
\(L(\frac12)\) \(\approx\) \(0.1454051105\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2 \( 1 - 3 T + 3 T^{2} - 3 T^{4} + 3 p T^{5} - 11 T^{6} + 3 p^{2} T^{7} - 3 p^{2} T^{8} + 3 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
5 \( ( 1 - 3 T + 3 p T^{2} - 27 T^{3} + 3 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
11 \( ( 1 + 6 T + 42 T^{2} + 135 T^{3} + 42 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 + 3 T + 3 T^{2} + 76 T^{3} + 45 T^{4} - 135 T^{5} + 3246 T^{6} - 135 p T^{7} + 45 p^{2} T^{8} + 76 p^{3} T^{9} + 3 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 6 T - 24 T^{2} - 54 T^{3} + 1338 T^{4} + 1914 T^{5} - 18929 T^{6} + 1914 p T^{7} + 1338 p^{2} T^{8} - 54 p^{3} T^{9} - 24 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 3 T - 42 T^{2} - 41 T^{3} + 1341 T^{4} + 216 T^{5} - 29541 T^{6} + 216 p T^{7} + 1341 p^{2} T^{8} - 41 p^{3} T^{9} - 42 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( ( 1 + 12 T + 96 T^{2} + 549 T^{3} + 96 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( 1 - 9 T + 30 T^{2} - 81 T^{3} - 579 T^{4} + 9414 T^{5} - 59051 T^{6} + 9414 p T^{7} - 579 p^{2} T^{8} - 81 p^{3} T^{9} + 30 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 3 T - 6 T^{2} + 319 T^{3} + 171 T^{4} - 1962 T^{5} + 62727 T^{6} - 1962 p T^{7} + 171 p^{2} T^{8} + 319 p^{3} T^{9} - 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 3 T - 24 T^{2} - 301 T^{3} + 171 T^{4} + 6552 T^{5} + 58893 T^{6} + 6552 p T^{7} + 171 p^{2} T^{8} - 301 p^{3} T^{9} - 24 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 114 T^{2} - 18 T^{3} + 8322 T^{4} + 1026 T^{5} - 394913 T^{6} + 1026 p T^{7} + 8322 p^{2} T^{8} - 18 p^{3} T^{9} - 114 p^{4} T^{10} + p^{6} T^{12} \)
43 \( 1 - 3 T - 114 T^{2} + 149 T^{3} + 9063 T^{4} - 5670 T^{5} - 441093 T^{6} - 5670 p T^{7} + 9063 p^{2} T^{8} + 149 p^{3} T^{9} - 114 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 3 T - 78 T^{2} - 405 T^{3} + 2481 T^{4} + 11064 T^{5} - 57089 T^{6} + 11064 p T^{7} + 2481 p^{2} T^{8} - 405 p^{3} T^{9} - 78 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 6 T - 114 T^{2} + 378 T^{3} + 10716 T^{4} - 17304 T^{5} - 587549 T^{6} - 17304 p T^{7} + 10716 p^{2} T^{8} + 378 p^{3} T^{9} - 114 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 3 T - 96 T^{2} + 495 T^{3} + 3615 T^{4} - 15798 T^{5} - 107021 T^{6} - 15798 p T^{7} + 3615 p^{2} T^{8} + 495 p^{3} T^{9} - 96 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 6 T - 132 T^{2} + 418 T^{3} + 13698 T^{4} - 19134 T^{5} - 893289 T^{6} - 19134 p T^{7} + 13698 p^{2} T^{8} + 418 p^{3} T^{9} - 132 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 12 T - 78 T^{2} + 518 T^{3} + 15318 T^{4} - 50094 T^{5} - 815637 T^{6} - 50094 p T^{7} + 15318 p^{2} T^{8} + 518 p^{3} T^{9} - 78 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 9 T + 159 T^{2} + 1305 T^{3} + 159 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 + 21 T + 138 T^{2} + 769 T^{3} + 10953 T^{4} + 30402 T^{5} - 450903 T^{6} + 30402 p T^{7} + 10953 p^{2} T^{8} + 769 p^{3} T^{9} + 138 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 21 T + 84 T^{2} - 499 T^{3} + 25767 T^{4} - 195678 T^{5} + 408327 T^{6} - 195678 p T^{7} + 25767 p^{2} T^{8} - 499 p^{3} T^{9} + 84 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 18 T + 30 T^{2} + 702 T^{3} + 8088 T^{4} - 126648 T^{5} + 719359 T^{6} - 126648 p T^{7} + 8088 p^{2} T^{8} + 702 p^{3} T^{9} + 30 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 12 T - 60 T^{2} - 198 T^{3} + 7584 T^{4} - 70800 T^{5} - 1684181 T^{6} - 70800 p T^{7} + 7584 p^{2} T^{8} - 198 p^{3} T^{9} - 60 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 3 T - 114 T^{2} - 149 T^{3} + 2421 T^{4} - 11502 T^{5} + 340233 T^{6} - 11502 p T^{7} + 2421 p^{2} T^{8} - 149 p^{3} T^{9} - 114 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.17303071797094609868885038061, −5.15366061712346839180186610648, −4.84385619798518983660869980935, −4.73160521895133367583547822920, −4.34081417999578037361961326784, −4.18676053613416602737986297921, −4.06113187401531617900241813433, −4.04508201086301766559115792359, −3.84770615001779251475183546967, −3.76996657511027708589574147073, −3.62857319485165329971567046821, −3.18482075722025928784951019407, −2.84734102705517232696322248630, −2.69524914056701787578109700420, −2.54897477651709362059081427405, −2.54120911041925421090821047615, −2.40439969192885239379211965645, −2.34599141543620026831377464676, −1.92621702527639761417071294844, −1.91511819307399926478080227636, −1.84944255140062579748557206187, −1.47726445250335300914774040503, −1.29352171036580509115759195581, −0.33015190191875693820466365416, −0.04401774381300624354247997699, 0.04401774381300624354247997699, 0.33015190191875693820466365416, 1.29352171036580509115759195581, 1.47726445250335300914774040503, 1.84944255140062579748557206187, 1.91511819307399926478080227636, 1.92621702527639761417071294844, 2.34599141543620026831377464676, 2.40439969192885239379211965645, 2.54120911041925421090821047615, 2.54897477651709362059081427405, 2.69524914056701787578109700420, 2.84734102705517232696322248630, 3.18482075722025928784951019407, 3.62857319485165329971567046821, 3.76996657511027708589574147073, 3.84770615001779251475183546967, 4.04508201086301766559115792359, 4.06113187401531617900241813433, 4.18676053613416602737986297921, 4.34081417999578037361961326784, 4.73160521895133367583547822920, 4.84385619798518983660869980935, 5.15366061712346839180186610648, 5.17303071797094609868885038061

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

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