Properties

Label 2-115920-1.1-c1-0-1
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 − 4·17-s + 23-s + 25-s − 6·29-s + 4·31-s − 35-s + 2·37-s − 4·41-s − 4·43-s − 8·47-s + 49-s + 10·53-s + 4·55-s − 10·59-s + 4·61-s + 12·67-s − 8·71-s + 2·73-s − 4·77-s + 2·79-s − 2·83-s + 4·85-s + 10·89-s + 14·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 1.20·11-s − 0.970·17-s + 0.208·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s − 0.169·35-s + 0.328·37-s − 0.624·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s + 1.37·53-s + 0.539·55-s − 1.30·59-s + 0.512·61-s + 1.46·67-s − 0.949·71-s + 0.234·73-s − 0.455·77-s + 0.225·79-s − 0.219·83-s + 0.433·85-s + 1.05·89-s + 1.42·97-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

\(L(1)\) \(\approx\) \(0.8340881387\)
\(L(\frac12)\) \(\approx\) \(0.8340881387\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−13.45142081264296, −13.14817130629944, −12.77411606398081, −12.08344598168186, −11.56834581943331, −11.25717553662252, −10.69135377661539, −10.28846001003696, −9.750932446807348, −9.099637881067495, −8.632693150290250, −8.093225344296573, −7.755656221917156, −7.174337966359071, −6.648901559638328, −6.085665279801587, −5.347736732313762, −4.978361356627779, −4.475558556540317, −3.799201929670744, −3.237156867221037, −2.509364517947844, −2.065045539295882, −1.224375176381363, −0.2857582999107749, 0.2857582999107749, 1.224375176381363, 2.065045539295882, 2.509364517947844, 3.237156867221037, 3.799201929670744, 4.475558556540317, 4.978361356627779, 5.347736732313762, 6.085665279801587, 6.648901559638328, 7.174337966359071, 7.755656221917156, 8.093225344296573, 8.632693150290250, 9.099637881067495, 9.750932446807348, 10.28846001003696, 10.69135377661539, 11.25717553662252, 11.56834581943331, 12.08344598168186, 12.77411606398081, 13.14817130629944, 13.45142081264296

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