Properties

Label 2-115920-1.1-c1-0-76
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 + 6·13-s + 6·17-s − 6·19-s − 23-s + 25-s + 2·29-s + 8·31-s − 35-s − 8·37-s + 10·41-s + 4·43-s − 12·47-s + 49-s + 6·53-s + 4·55-s + 10·59-s + 6·61-s + 6·65-s + 12·67-s − 2·71-s + 6·73-s − 4·77-s + 2·79-s + 4·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 1.20·11-s + 1.66·13-s + 1.45·17-s − 1.37·19-s − 0.208·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.169·35-s − 1.31·37-s + 1.56·41-s + 0.609·43-s − 1.75·47-s + 1/7·49-s + 0.824·53-s + 0.539·55-s + 1.30·59-s + 0.768·61-s + 0.744·65-s + 1.46·67-s − 0.237·71-s + 0.702·73-s − 0.455·77-s + 0.225·79-s + 0.439·83-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

\(L(1)\) \(\approx\) \(4.338023718\)
\(L(\frac12)\) \(\approx\) \(4.338023718\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−13.59987393310650, −13.13981526772653, −12.65214539278725, −12.18564665405070, −11.68106932294607, −11.18202813117249, −10.66142376101919, −10.10243587354054, −9.809774931664962, −9.139059345831409, −8.633343127073512, −8.346472212515611, −7.740152611650026, −6.906592308591180, −6.458757216977368, −6.191633440357163, −5.672478529571531, −4.986510960692274, −4.270245953788664, −3.651948062980905, −3.479456133453510, −2.528525433227388, −1.905733958578325, −1.109775334910532, −0.7576964701976165, 0.7576964701976165, 1.109775334910532, 1.905733958578325, 2.528525433227388, 3.479456133453510, 3.651948062980905, 4.270245953788664, 4.986510960692274, 5.672478529571531, 6.191633440357163, 6.458757216977368, 6.906592308591180, 7.740152611650026, 8.346472212515611, 8.633343127073512, 9.139059345831409, 9.809774931664962, 10.10243587354054, 10.66142376101919, 11.18202813117249, 11.68106932294607, 12.18564665405070, 12.65214539278725, 13.13981526772653, 13.59987393310650

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