Properties

Label 2-115920-1.1-c1-0-103
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 7-s − 6·11-s + 4·13-s − 2·17-s + 4·19-s + 23-s + 25-s − 6·29-s + 4·31-s + 35-s + 4·37-s + 2·41-s + 6·43-s + 49-s − 8·53-s − 6·55-s + 6·59-s + 6·61-s + 4·65-s − 10·67-s − 14·73-s − 6·77-s − 16·79-s − 4·83-s − 2·85-s + 2·89-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 1.80·11-s + 1.10·13-s − 0.485·17-s + 0.917·19-s + 0.208·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s + 0.169·35-s + 0.657·37-s + 0.312·41-s + 0.914·43-s + 1/7·49-s − 1.09·53-s − 0.809·55-s + 0.781·59-s + 0.768·61-s + 0.496·65-s − 1.22·67-s − 1.63·73-s − 0.683·77-s − 1.80·79-s − 0.439·83-s − 0.216·85-s + 0.211·89-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: \(1\)
Selberg data: \((2,\ 115920,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−13.58082661115553, −13.39808277613845, −13.04881374066887, −12.54199942390155, −11.82554159011037, −11.29168538277083, −10.92819053061015, −10.53979742287824, −9.937307615753104, −9.506614022215970, −8.902584709075085, −8.355825724857111, −7.984093666522333, −7.376431280746068, −7.010339821064366, −6.143099147784556, −5.635031328796104, −5.485690883370539, −4.599460811022078, −4.297415287341843, −3.350032202175369, −2.876603111507409, −2.321311978413112, −1.593497921977047, −0.9225562489856388, 0, 0.9225562489856388, 1.593497921977047, 2.321311978413112, 2.876603111507409, 3.350032202175369, 4.297415287341843, 4.599460811022078, 5.485690883370539, 5.635031328796104, 6.143099147784556, 7.010339821064366, 7.376431280746068, 7.984093666522333, 8.355825724857111, 8.902584709075085, 9.506614022215970, 9.937307615753104, 10.53979742287824, 10.92819053061015, 11.29168538277083, 11.82554159011037, 12.54199942390155, 13.04881374066887, 13.39808277613845, 13.58082661115553

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