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Properties

Label 12-2015e6-1.1-c0e6-0-2
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

Downloads

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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

\(L(\frac{1}{2})\) \(\approx\) \(0.5643875915\)
\(L(\frac12)\) \(\approx\) \(0.5643875915\)
\(L(1)\) 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

−4.81710240780224684634693887130, −4.75446753065031788836569066579, −4.68305682981346806552717712588, −4.50394950790259513652408786338, −4.36879884019262680496021940926, −4.22694667553507015242393324442, −4.19273826275226385932867657323, −4.13240535960513333598165367327, −3.87673037531629597240929452920, −3.59290507125894400296651150092, −3.40612690067797073059925566503, −3.32605393727601151168199851474, −3.02315865078478646114561752633, −2.95804221728389852285614058661, −2.77709517962760665606276343950, −2.76464103271535897070692842725, −2.67526900113930576761134083982, −2.41031060249119668675565636561, −2.32374434277634867372010009746, −2.20353432249713595596502124490, −1.40305261835672187330959121475, −1.28824742785316489609918592529, −0.812816826684260909474368882356, −0.62218055990875832448441706626, −0.52864828207844568732104807898, 0.52864828207844568732104807898, 0.62218055990875832448441706626, 0.812816826684260909474368882356, 1.28824742785316489609918592529, 1.40305261835672187330959121475, 2.20353432249713595596502124490, 2.32374434277634867372010009746, 2.41031060249119668675565636561, 2.67526900113930576761134083982, 2.76464103271535897070692842725, 2.77709517962760665606276343950, 2.95804221728389852285614058661, 3.02315865078478646114561752633, 3.32605393727601151168199851474, 3.40612690067797073059925566503, 3.59290507125894400296651150092, 3.87673037531629597240929452920, 4.13240535960513333598165367327, 4.19273826275226385932867657323, 4.22694667553507015242393324442, 4.36879884019262680496021940926, 4.50394950790259513652408786338, 4.68305682981346806552717712588, 4.75446753065031788836569066579, 4.81710240780224684634693887130

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

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