Properties

Label 2-8112-1.1-c1-0-20
Degree $2$
Conductor $8112$
Sign $1$
Analytic cond. $64.7746$
Root an. cond. $8.04826$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3·5-s − 2·7-s + 9-s − 6·11-s − 3·15-s − 3·17-s − 2·19-s + 2·21-s + 6·23-s + 4·25-s − 27-s + 3·29-s + 4·31-s + 6·33-s − 6·35-s − 7·37-s − 3·41-s + 10·43-s + 3·45-s − 6·47-s − 3·49-s + 3·51-s + 3·53-s − 18·55-s + 2·57-s − 7·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s − 0.755·7-s + 1/3·9-s − 1.80·11-s − 0.774·15-s − 0.727·17-s − 0.458·19-s + 0.436·21-s + 1.25·23-s + 4/5·25-s − 0.192·27-s + 0.557·29-s + 0.718·31-s + 1.04·33-s − 1.01·35-s − 1.15·37-s − 0.468·41-s + 1.52·43-s + 0.447·45-s − 0.875·47-s − 3/7·49-s + 0.420·51-s + 0.412·53-s − 2.42·55-s + 0.264·57-s − 0.896·61-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8112,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.353557831\)
\(L(\frac12)\) \(\approx\) \(1.353557831\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−7.71228890286374164796051264470, −6.94066222555462250579818106752, −6.33681695871009114332207145716, −5.81439251540901211398105316257, −5.06733563554819733912021363064, −4.65047702520769116257093010231, −3.28311515952908152854941075889, −2.59687322896259811851599454306, −1.87740828212772298640276912541, −0.56719806990473160362827845912, 0.56719806990473160362827845912, 1.87740828212772298640276912541, 2.59687322896259811851599454306, 3.28311515952908152854941075889, 4.65047702520769116257093010231, 5.06733563554819733912021363064, 5.81439251540901211398105316257, 6.33681695871009114332207145716, 6.94066222555462250579818106752, 7.71228890286374164796051264470

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