Properties

Label 2-115920-1.1-c1-0-25
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 − 2·17-s + 4·19-s − 23-s + 25-s + 2·29-s − 8·31-s − 35-s + 6·37-s − 2·41-s − 4·43-s + 8·47-s + 49-s + 2·53-s − 4·55-s − 4·59-s + 14·61-s + 2·65-s + 4·67-s + 2·73-s + 4·77-s + 12·83-s + 2·85-s + 6·89-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s + 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s − 0.169·35-s + 0.986·37-s − 0.312·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.274·53-s − 0.539·55-s − 0.520·59-s + 1.79·61-s + 0.248·65-s + 0.488·67-s + 0.234·73-s + 0.455·77-s + 1.31·83-s + 0.216·85-s + 0.635·89-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

\(L(1)\) \(\approx\) \(2.585240110\)
\(L(\frac12)\) \(\approx\) \(2.585240110\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−13.63797518051843, −13.07929001832275, −12.61479932196556, −11.98657598443403, −11.67367537960404, −11.37501274119261, −10.69959209769898, −10.25352230812036, −9.529260916755263, −9.238300805346889, −8.734997762938494, −8.148592618637886, −7.578561848145121, −7.194924008054234, −6.653686740840222, −6.116948757608631, −5.432354290817660, −4.933904510552528, −4.374397416773660, −3.749831975276111, −3.422226896473943, −2.489134169894878, −1.975179790693010, −1.165456017008614, −0.5422817550867035, 0.5422817550867035, 1.165456017008614, 1.975179790693010, 2.489134169894878, 3.422226896473943, 3.749831975276111, 4.374397416773660, 4.933904510552528, 5.432354290817660, 6.116948757608631, 6.653686740840222, 7.194924008054234, 7.578561848145121, 8.148592618637886, 8.734997762938494, 9.238300805346889, 9.529260916755263, 10.25352230812036, 10.69959209769898, 11.37501274119261, 11.67367537960404, 11.98657598443403, 12.61479932196556, 13.07929001832275, 13.63797518051843

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