Properties

Label 2-115920-1.1-c1-0-12
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 7-s + 4·11-s + 2·13-s − 4·17-s + 4·19-s + 23-s + 25-s + 8·29-s − 8·31-s + 35-s − 2·37-s − 4·41-s + 8·43-s + 49-s + 4·53-s − 4·55-s − 8·59-s − 2·61-s − 2·65-s − 16·67-s − 4·71-s − 4·77-s + 4·79-s − 12·83-s + 4·85-s − 6·89-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s + 1.20·11-s + 0.554·13-s − 0.970·17-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 − 0.328·37-s − 0.624·41-s + 1.21·43-s + 1/7·49-s + 0.549·53-s − 0.539·55-s − 1.04·59-s − 0.256·61-s − 0.248·65-s − 1.95·67-s − 0.474·71-s − 0.455·77-s + 0.450·79-s − 1.31·83-s + 0.433·85-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

\(L(1)\) \(\approx\) \(1.983661368\)
\(L(\frac12)\) \(\approx\) \(1.983661368\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.55824792049493, −13.24078604294820, −12.45031758910372, −12.19832408817134, −11.71723986405176, −11.11732867641231, −10.85653222282766, −10.19378763407679, −9.623275707407170, −9.046563337302701, −8.819603088694760, −8.296927580905197, −7.473538166085313, −7.158002376357972, −6.622171994756456, −6.113779997395453, −5.611107129880105, −4.853840436713948, −4.268460738009860, −3.898778431845396, −3.182022455579974, −2.793068712562924, −1.778691979012005, −1.261989604608226, −0.4576378555286784, 0.4576378555286784, 1.261989604608226, 1.778691979012005, 2.793068712562924, 3.182022455579974, 3.898778431845396, 4.268460738009860, 4.853840436713948, 5.611107129880105, 6.113779997395453, 6.622171994756456, 7.158002376357972, 7.473538166085313, 8.296927580905197, 8.819603088694760, 9.046563337302701, 9.623275707407170, 10.19378763407679, 10.85653222282766, 11.11732867641231, 11.71723986405176, 12.19832408817134, 12.45031758910372, 13.24078604294820, 13.55824792049493

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