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Properties

Label 2-115920-1.1-c1-0-49
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 + 11-s − 2·17-s + 3·19-s + 23-s + 25-s + 6·29-s + 35-s + 6·37-s + 3·41-s + 6·43-s − 7·47-s + 49-s + 7·53-s + 55-s − 3·59-s + 61-s + 2·67-s + 12·71-s − 14·73-s + 77-s + 10·79-s + 4·83-s − 2·85-s + 8·89-s + 3·95-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 0.301·11-s − 0.485·17-s + 0.688·19-s + 0.208·23-s + 1/5·25-s + 1.11·29-s + 0.169·35-s + 0.986·37-s + 0.468·41-s + 0.914·43-s − 1.02·47-s + 1/7·49-s + 0.961·53-s + 0.134·55-s − 0.390·59-s + 0.128·61-s + 0.244·67-s + 1.42·71-s − 1.63·73-s + 0.113·77-s + 1.12·79-s + 0.439·83-s − 0.216·85-s + 0.847·89-s + 0.307·95-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 115920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.606702235\)
\(L(\frac12)\) \(\approx\) \(3.606702235\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 7 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.52552646840810, −13.24531928733074, −12.64057248535967, −12.12998801717482, −11.64991165757957, −11.20144655170521, −10.69363157094945, −10.19343393599813, −9.671781632121850, −9.158071790767983, −8.809601906937362, −8.102115656750014, −7.715400785640050, −7.121241484625253, −6.446047732315059, −6.226191693638671, −5.385959407376019, −5.064300743357833, −4.365335963692306, −3.929805096303634, −3.087116903321729, −2.592643194323945, −1.954146637195173, −1.195429675129881, −0.6360175737342446, 0.6360175737342446, 1.195429675129881, 1.954146637195173, 2.592643194323945, 3.087116903321729, 3.929805096303634, 4.365335963692306, 5.064300743357833, 5.385959407376019, 6.226191693638671, 6.446047732315059, 7.121241484625253, 7.715400785640050, 8.102115656750014, 8.809601906937362, 9.158071790767983, 9.671781632121850, 10.19343393599813, 10.69363157094945, 11.20144655170521, 11.64991165757957, 12.12998801717482, 12.64057248535967, 13.24531928733074, 13.52552646840810

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