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Properties

Label 12-2015e6-1.1-c0e6-0-0
Degree $12$
Conductor $6.693\times 10^{19}$
Sign $1$
Analytic cond. $1.03417$
Root an. cond. $1.00280$
Motivic weight $0$
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  + 2-s − 3-s − 6·5-s − 6-s + 7-s − 6·10-s + 11-s + 6·13-s + 14-s + 6·15-s − 17-s − 21-s + 22-s − 23-s + 21·25-s + 6·26-s + 6·30-s − 6·31-s − 33-s − 34-s − 6·35-s − 6·39-s − 42-s − 43-s − 46-s + 47-s + 21·50-s + ⋯
L(s)  = 1  + 2-s − 3-s − 6·5-s − 6-s + 7-s − 6·10-s + 11-s + 6·13-s + 14-s + 6·15-s − 17-s − 21-s + 22-s − 23-s + 21·25-s + 6·26-s + 6·30-s − 6·31-s − 33-s − 34-s − 6·35-s − 6·39-s − 42-s − 43-s − 46-s + 47-s + 21·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 13^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 13^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(5^{6} \cdot 13^{6} \cdot 31^{6}\)
Sign: $1$
Analytic conductor: \(1.03417\)
Root analytic conductor: \(1.00280\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 5^{6} \cdot 13^{6} \cdot 31^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3313154138\)
\(L(\frac12)\) \(\approx\) \(0.3313154138\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( ( 1 + T )^{6} \)
13 \( ( 1 - T )^{6} \)
31 \( ( 1 + T )^{6} \)
good2 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} \)
3 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
7 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} \)
11 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} \)
17 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
19 \( ( 1 - T )^{6}( 1 + T )^{6} \)
23 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
29 \( ( 1 - T )^{6}( 1 + T )^{6} \)
37 \( ( 1 - T )^{6}( 1 + T )^{6} \)
41 \( ( 1 - T )^{6}( 1 + T )^{6} \)
43 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
47 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} \)
53 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
59 \( ( 1 - T )^{6}( 1 + T )^{6} \)
61 \( ( 1 - T )^{6}( 1 + T )^{6} \)
67 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} \)
71 \( ( 1 - T )^{6}( 1 + T )^{6} \)
73 \( ( 1 - T )^{6}( 1 + T )^{6} \)
79 \( ( 1 - T )^{6}( 1 + T )^{6} \)
83 \( ( 1 - T )^{6}( 1 + T )^{6} \)
89 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} \)
97 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} \)
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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

−4.84581773209251315981198053136, −4.79343622931051103821214072591, −4.53252905307959028205511788791, −4.33237265664693435258092036523, −4.30112016051150424567699631710, −4.12437415305407235023098340586, −4.06606629102306118297321111963, −3.92253042532431160074776311965, −3.73007221758697957997331910277, −3.66434342184411936467986469188, −3.64885746401840879835378770179, −3.55247825461183670086020724073, −3.38615276817347851932622009752, −3.30138835769471909622945047789, −3.01170488061848822352420904981, −2.79131823306619987267679171894, −2.66675529416695072123998211271, −2.01588216875098740246716907165, −1.90786869900449721619756199059, −1.64585215054058414052770944440, −1.51629558898815477493012425368, −1.29692522388745557124190793142, −0.960778428313051497684955063666, −0.78841751567386737534281202557, −0.35048341944976075604806210579, 0.35048341944976075604806210579, 0.78841751567386737534281202557, 0.960778428313051497684955063666, 1.29692522388745557124190793142, 1.51629558898815477493012425368, 1.64585215054058414052770944440, 1.90786869900449721619756199059, 2.01588216875098740246716907165, 2.66675529416695072123998211271, 2.79131823306619987267679171894, 3.01170488061848822352420904981, 3.30138835769471909622945047789, 3.38615276817347851932622009752, 3.55247825461183670086020724073, 3.64885746401840879835378770179, 3.66434342184411936467986469188, 3.73007221758697957997331910277, 3.92253042532431160074776311965, 4.06606629102306118297321111963, 4.12437415305407235023098340586, 4.30112016051150424567699631710, 4.33237265664693435258092036523, 4.53252905307959028205511788791, 4.79343622931051103821214072591, 4.84581773209251315981198053136

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

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