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Properties

Label 12-3024e6-1.1-c1e6-0-0
Degree $12$
Conductor $7.647\times 10^{20}$
Sign $1$
Analytic cond. $1.98222\times 10^{8}$
Root an. cond. $4.91393$
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  + 5-s − 3·7-s − 2·11-s − 3·13-s − 4·17-s − 6·19-s − 14·23-s + 11·25-s + 29-s − 3·31-s − 3·35-s − 6·37-s + 3·43-s − 21·47-s + 3·49-s − 12·53-s − 2·55-s − 31·59-s − 6·61-s − 3·65-s + 6·67-s + 34·71-s − 6·73-s + 6·77-s − 9·79-s − 20·83-s − 4·85-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.13·7-s − 0.603·11-s − 0.832·13-s − 0.970·17-s − 1.37·19-s − 2.91·23-s + 11/5·25-s + 0.185·29-s − 0.538·31-s − 0.507·35-s − 0.986·37-s + 0.457·43-s − 3.06·47-s + 3/7·49-s − 1.64·53-s − 0.269·55-s − 4.03·59-s − 0.768·61-s − 0.372·65-s + 0.733·67-s + 4.03·71-s − 0.702·73-s + 0.683·77-s − 1.01·79-s − 2.19·83-s − 0.433·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{18} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(1.98222\times 10^{8}\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{18} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.04606862151\)
\(L(\frac12)\) \(\approx\) \(0.04606862151\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( ( 1 + T + T^{2} )^{3} \)
good5 \( 1 - T - 2 p T^{2} + 7 T^{3} + 57 T^{4} - 14 T^{5} - 299 T^{6} - 14 p T^{7} + 57 p^{2} T^{8} + 7 p^{3} T^{9} - 2 p^{5} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 2 T - 4 T^{2} + 46 T^{3} + 6 T^{4} - 230 T^{5} + 1699 T^{6} - 230 p T^{7} + 6 p^{2} T^{8} + 46 p^{3} T^{9} - 4 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 3 T + 3 T^{2} - 84 T^{3} - 15 p T^{4} + 345 T^{5} + 5006 T^{6} + 345 p T^{7} - 15 p^{3} T^{8} - 84 p^{3} T^{9} + 3 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( ( 1 + 2 T + 32 T^{2} + 21 T^{3} + 32 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( ( 1 + 3 T + 33 T^{2} + 35 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 + 14 T + 74 T^{2} + 358 T^{3} + 2628 T^{4} + 11188 T^{5} + 33943 T^{6} + 11188 p T^{7} + 2628 p^{2} T^{8} + 358 p^{3} T^{9} + 74 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - T - 46 T^{2} - 149 T^{3} + 897 T^{4} + 4282 T^{5} - 13523 T^{6} + 4282 p T^{7} + 897 p^{2} T^{8} - 149 p^{3} T^{9} - 46 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 3 T - 48 T^{2} - 147 T^{3} + 1005 T^{4} + 1344 T^{5} - 24505 T^{6} + 1344 p T^{7} + 1005 p^{2} T^{8} - 147 p^{3} T^{9} - 48 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + 3 T + 81 T^{2} + 199 T^{3} + 81 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 - 90 T^{2} - 18 T^{3} + 4410 T^{4} + 810 T^{5} - 194177 T^{6} + 810 p T^{7} + 4410 p^{2} T^{8} - 18 p^{3} T^{9} - 90 p^{4} T^{10} + p^{6} T^{12} \)
43 \( 1 - 3 T - 24 T^{2} + 979 T^{3} - 1947 T^{4} - 14820 T^{5} + 386067 T^{6} - 14820 p T^{7} - 1947 p^{2} T^{8} + 979 p^{3} T^{9} - 24 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 21 T + 180 T^{2} + 1119 T^{3} + 10053 T^{4} + 100416 T^{5} + 788551 T^{6} + 100416 p T^{7} + 10053 p^{2} T^{8} + 1119 p^{3} T^{9} + 180 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 + 6 T + 162 T^{2} + 627 T^{3} + 162 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 + 31 T + 476 T^{2} + 5741 T^{3} + 62553 T^{4} + 587576 T^{5} + 4781851 T^{6} + 587576 p T^{7} + 62553 p^{2} T^{8} + 5741 p^{3} T^{9} + 476 p^{4} T^{10} + 31 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 6 T + 48 T^{2} + 642 T^{3} + 3018 T^{4} + 35394 T^{5} + 438671 T^{6} + 35394 p T^{7} + 3018 p^{2} T^{8} + 642 p^{3} T^{9} + 48 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 6 T - 150 T^{2} + 506 T^{3} + 17268 T^{4} - 28236 T^{5} - 1220289 T^{6} - 28236 p T^{7} + 17268 p^{2} T^{8} + 506 p^{3} T^{9} - 150 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 - 17 T + 119 T^{2} - 507 T^{3} + 119 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 + 3 T + 195 T^{2} + 359 T^{3} + 195 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( 1 + 9 T - 114 T^{2} - 351 T^{3} + 13143 T^{4} - 15786 T^{5} - 1414609 T^{6} - 15786 p T^{7} + 13143 p^{2} T^{8} - 351 p^{3} T^{9} - 114 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 20 T + 38 T^{2} + 346 T^{3} + 32058 T^{4} + 183754 T^{5} - 606869 T^{6} + 183754 p T^{7} + 32058 p^{2} T^{8} + 346 p^{3} T^{9} + 38 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \)
89 \( ( 1 + 12 T + 216 T^{2} + 1425 T^{3} + 216 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 9 T - 66 T^{2} + 2023 T^{3} - 7707 T^{4} - 73950 T^{5} + 1766073 T^{6} - 73950 p T^{7} - 7707 p^{2} T^{8} + 2023 p^{3} T^{9} - 66 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.55722254406702205510854809435, −4.38840424291483842304071451171, −4.33534996739385558363689224654, −4.14135768094899534199932360534, −4.07833611837266544828786851190, −3.51756582843446544938489524001, −3.44139948088772122480365129184, −3.41653623758742040770993786458, −3.37679737026773433946573289190, −3.35925284271521091340382240818, −3.03508962188100466643556940719, −2.83948157343850768225969440704, −2.64151666004063412786011658146, −2.48478581264795878982577228999, −2.28056531545647835145056872750, −2.15518567017538656927402285057, −2.01637869856489855829989560073, −1.97016295183801251939052187975, −1.66733414498183871787746634501, −1.36092471068714041722190054377, −1.26177971472684245819120758954, −1.14163614234246828816040476494, −0.55203204299589915481174572021, −0.16428096490706031055429452358, −0.07703309323769526066293749891, 0.07703309323769526066293749891, 0.16428096490706031055429452358, 0.55203204299589915481174572021, 1.14163614234246828816040476494, 1.26177971472684245819120758954, 1.36092471068714041722190054377, 1.66733414498183871787746634501, 1.97016295183801251939052187975, 2.01637869856489855829989560073, 2.15518567017538656927402285057, 2.28056531545647835145056872750, 2.48478581264795878982577228999, 2.64151666004063412786011658146, 2.83948157343850768225969440704, 3.03508962188100466643556940719, 3.35925284271521091340382240818, 3.37679737026773433946573289190, 3.41653623758742040770993786458, 3.44139948088772122480365129184, 3.51756582843446544938489524001, 4.07833611837266544828786851190, 4.14135768094899534199932360534, 4.33534996739385558363689224654, 4.38840424291483842304071451171, 4.55722254406702205510854809435

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

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