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Properties

Label 12-1323e6-1.1-c1e6-0-9
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 − 3·5-s + 9·8-s − 9·10-s + 6·11-s − 3·13-s + 12·16-s + 12·17-s + 6·19-s − 18·20-s + 18·22-s + 12·23-s + 15·25-s − 9·26-s + 9·29-s − 3·31-s + 12·32-s + 36·34-s − 6·37-s + 18·38-s − 27·40-s + 3·43-s + 36·44-s + 36·46-s − 3·47-s + 45·50-s + ⋯
L(s)  = 1  + 2.12·2-s + 3·4-s − 1.34·5-s + 3.18·8-s − 2.84·10-s + 1.80·11-s − 0.832·13-s + 3·16-s + 2.91·17-s + 1.37·19-s − 4.02·20-s + 3.83·22-s + 2.50·23-s + 3·25-s − 1.76·26-s + 1.67·29-s − 0.538·31-s + 2.12·32-s + 6.17·34-s − 0.986·37-s + 2.91·38-s − 4.26·40-s + 0.457·43-s + 5.42·44-s + 5.30·46-s − 0.437·47-s + 6.36·50-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

\(L(1)\) \(\approx\) \(52.49124489\)
\(L(\frac12)\) \(\approx\) \(52.49124489\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T - 6 T^{2} - 9 T^{3} + 69 T^{4} + 6 p T^{5} - 371 T^{6} + 6 p^{2} T^{7} + 69 p^{2} T^{8} - 9 p^{3} T^{9} - 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 6 T - 6 T^{2} + 18 T^{3} + 492 T^{4} - 852 T^{5} - 2873 T^{6} - 852 p T^{7} + 492 p^{2} T^{8} + 18 p^{3} T^{9} - 6 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
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 \( ( 1 - 6 T + 60 T^{2} - 207 T^{3} + 60 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( ( 1 - 3 T + 51 T^{2} - 97 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 12 T + 48 T^{2} - 54 T^{3} + 420 T^{4} - 6060 T^{5} + 37591 T^{6} - 6060 p T^{7} + 420 p^{2} T^{8} - 54 p^{3} T^{9} + 48 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
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 \( ( 1 + 3 T + 33 T^{2} - 101 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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 \( ( 1 + 6 T + 150 T^{2} + 639 T^{3} + 150 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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 \( ( 1 - 21 T + 303 T^{2} - 2797 T^{3} + 303 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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 \( ( 1 - 12 T + 204 T^{2} - 1323 T^{3} + 204 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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

−4.95973074714213701096764667052, −4.86337531250455061393703889378, −4.79147089820735622507406293004, −4.74002776479727225896270540495, −4.52871536025755474280684628405, −4.37253054214197540511969216244, −4.12577086842584785586091838826, −3.66678144417952125600788567849, −3.58286927243775508975243557360, −3.56104949451016729577983807001, −3.49853898510576492463149524299, −3.49083068466768324086974592622, −3.09834158108941784834272591317, −3.07201088240351611640967905629, −3.06324282027410608708237589710, −2.73743623672108627455712007396, −2.28902996215265095176585872028, −2.13704803445286416651960842011, −1.99621321358078600021601484503, −1.97554423386622855880003377982, −1.29628801171628561929671616854, −1.02820778668868025514599698960, −0.944908187501929559442072601944, −0.903057635039349287963675294139, −0.66575441700325847777157285528, 0.66575441700325847777157285528, 0.903057635039349287963675294139, 0.944908187501929559442072601944, 1.02820778668868025514599698960, 1.29628801171628561929671616854, 1.97554423386622855880003377982, 1.99621321358078600021601484503, 2.13704803445286416651960842011, 2.28902996215265095176585872028, 2.73743623672108627455712007396, 3.06324282027410608708237589710, 3.07201088240351611640967905629, 3.09834158108941784834272591317, 3.49083068466768324086974592622, 3.49853898510576492463149524299, 3.56104949451016729577983807001, 3.58286927243775508975243557360, 3.66678144417952125600788567849, 4.12577086842584785586091838826, 4.37253054214197540511969216244, 4.52871536025755474280684628405, 4.74002776479727225896270540495, 4.79147089820735622507406293004, 4.86337531250455061393703889378, 4.95973074714213701096764667052

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

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