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Properties

Label 2-115920-1.1-c1-0-4
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 − 2·13-s − 2·17-s + 4·19-s − 23-s + 25-s + 2·29-s − 8·31-s + 35-s + 6·37-s + 6·41-s + 4·43-s + 8·47-s + 49-s + 10·53-s + 4·55-s − 12·59-s + 14·61-s + 2·65-s + 12·67-s + 16·71-s − 14·73-s + 4·77-s − 8·79-s − 12·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s + 0.169·35-s + 0.986·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s + 1.37·53-s + 0.539·55-s − 1.56·59-s + 1.79·61-s + 0.248·65-s + 1.46·67-s + 1.89·71-s − 1.63·73-s + 0.455·77-s − 0.900·79-s − 1.31·83-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 115920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.277090786\)
\(L(\frac12)\) \(\approx\) \(1.277090786\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−13.43276534056133, −13.17125815383199, −12.54211717388618, −12.32943673488169, −11.61764819925861, −11.16240864521572, −10.71918858762156, −10.22371241739862, −9.662557604257663, −9.247852735761090, −8.707686747636761, −8.000965639463686, −7.700394723407803, −7.148273231690506, −6.788132545477622, −5.885891509981201, −5.518364045935115, −5.053307195296760, −4.273730354022927, −3.916058579601965, −3.126630317533987, −2.565607092742024, −2.179440721115734, −1.064573719217186, −0.3871244981393732, 0.3871244981393732, 1.064573719217186, 2.179440721115734, 2.565607092742024, 3.126630317533987, 3.916058579601965, 4.273730354022927, 5.053307195296760, 5.518364045935115, 5.885891509981201, 6.788132545477622, 7.148273231690506, 7.700394723407803, 8.000965639463686, 8.707686747636761, 9.247852735761090, 9.662557604257663, 10.22371241739862, 10.71918858762156, 11.16240864521572, 11.61764819925861, 12.32943673488169, 12.54211717388618, 13.17125815383199, 13.43276534056133

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