Properties

Label 2-115920-1.1-c1-0-18
Degree $2$
Conductor $115920$
Sign $1$
Analytic cond. $925.625$
Root an. cond. $30.4240$
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  + 5-s − 7-s − 4·11-s − 2·13-s + 4·19-s − 23-s + 25-s − 8·29-s + 8·31-s − 35-s + 6·37-s + 8·41-s + 8·43-s + 49-s + 12·53-s − 4·55-s − 4·59-s + 10·61-s − 2·65-s + 12·71-s + 16·73-s + 4·77-s − 16·79-s − 16·83-s + 6·89-s + 2·91-s + 4·95-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 1.20·11-s − 0.554·13-s + 0.917·19-s − 0.208·23-s + 1/5·25-s − 1.48·29-s + 1.43·31-s − 0.169·35-s + 0.986·37-s + 1.24·41-s + 1.21·43-s + 1/7·49-s + 1.64·53-s − 0.539·55-s − 0.520·59-s + 1.28·61-s − 0.248·65-s + 1.42·71-s + 1.87·73-s + 0.455·77-s − 1.80·79-s − 1.75·83-s + 0.635·89-s + 0.209·91-s + 0.410·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115920\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(925.625\)
Root analytic conductor: \(30.4240\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{115920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 115920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.251278377\)
\(L(\frac12)\) \(\approx\) \(2.251278377\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
7 \( 1 + T \)
23 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.67612787516596, −12.99692012972107, −12.76939788091371, −12.28700325720214, −11.59383781726432, −11.17336645903389, −10.68927289723635, −9.981374539760947, −9.819660012964778, −9.338702400779625, −8.705748657804236, −8.084879231814397, −7.590488032010604, −7.258467545819476, −6.585511701912470, −5.921596047460809, −5.519624884492422, −5.132420916015181, −4.352028157660945, −3.879100685221431, −3.033266672058264, −2.526312777669142, −2.197640782597415, −1.113546946545742, −0.5027675478482009, 0.5027675478482009, 1.113546946545742, 2.197640782597415, 2.526312777669142, 3.033266672058264, 3.879100685221431, 4.352028157660945, 5.132420916015181, 5.519624884492422, 5.921596047460809, 6.585511701912470, 7.258467545819476, 7.590488032010604, 8.084879231814397, 8.705748657804236, 9.338702400779625, 9.819660012964778, 9.981374539760947, 10.68927289723635, 11.17336645903389, 11.59383781726432, 12.28700325720214, 12.76939788091371, 12.99692012972107, 13.67612787516596

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