Properties

Label 2-8112-1.1-c1-0-109
Degree $2$
Conductor $8112$
Sign $1$
Analytic cond. $64.7746$
Root an. cond. $8.04826$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 3.69·5-s + 0.801·7-s + 9-s + 2.85·11-s + 3.69·15-s + 2.93·17-s + 2.44·19-s + 0.801·21-s + 7.78·23-s + 8.63·25-s + 27-s + 3.85·29-s − 2.34·31-s + 2.85·33-s + 2.96·35-s + 7.44·37-s − 0.850·41-s + 1.61·43-s + 3.69·45-s − 2.44·47-s − 6.35·49-s + 2.93·51-s − 9.96·53-s + 10.5·55-s + 2.44·57-s + 5.38·59-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.65·5-s + 0.303·7-s + 0.333·9-s + 0.859·11-s + 0.953·15-s + 0.712·17-s + 0.560·19-s + 0.174·21-s + 1.62·23-s + 1.72·25-s + 0.192·27-s + 0.715·29-s − 0.421·31-s + 0.496·33-s + 0.500·35-s + 1.22·37-s − 0.132·41-s + 0.246·43-s + 0.550·45-s − 0.356·47-s − 0.908·49-s + 0.411·51-s − 1.36·53-s + 1.41·55-s + 0.323·57-s + 0.700·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8112\)    =    \(2^{4} \cdot 3 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(64.7746\)
Root analytic conductor: \(8.04826\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8112,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.653813892\)
\(L(\frac12)\) \(\approx\) \(4.653813892\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 \)
good5 \( 1 - 3.69T + 5T^{2} \)
7 \( 1 - 0.801T + 7T^{2} \)
11 \( 1 - 2.85T + 11T^{2} \)
17 \( 1 - 2.93T + 17T^{2} \)
19 \( 1 - 2.44T + 19T^{2} \)
23 \( 1 - 7.78T + 23T^{2} \)
29 \( 1 - 3.85T + 29T^{2} \)
31 \( 1 + 2.34T + 31T^{2} \)
37 \( 1 - 7.44T + 37T^{2} \)
41 \( 1 + 0.850T + 41T^{2} \)
43 \( 1 - 1.61T + 43T^{2} \)
47 \( 1 + 2.44T + 47T^{2} \)
53 \( 1 + 9.96T + 53T^{2} \)
59 \( 1 - 5.38T + 59T^{2} \)
61 \( 1 + 13.2T + 61T^{2} \)
67 \( 1 + 14.3T + 67T^{2} \)
71 \( 1 + 8.12T + 71T^{2} \)
73 \( 1 + 11.8T + 73T^{2} \)
79 \( 1 + 5.40T + 79T^{2} \)
83 \( 1 + 7.04T + 83T^{2} \)
89 \( 1 - 1.13T + 89T^{2} \)
97 \( 1 + 5.94T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.77466260867425778780909598637, −7.13787979668869273528889797176, −6.35182893571542793039384006329, −5.84994777525368762836324100569, −5.02392100430033571026840125576, −4.41902091258980306782709716996, −3.17633568766860780801573686458, −2.77503469413040750294299654414, −1.58879115433554025117382291857, −1.23792634125153490387343651101, 1.23792634125153490387343651101, 1.58879115433554025117382291857, 2.77503469413040750294299654414, 3.17633568766860780801573686458, 4.41902091258980306782709716996, 5.02392100430033571026840125576, 5.84994777525368762836324100569, 6.35182893571542793039384006329, 7.13787979668869273528889797176, 7.77466260867425778780909598637

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