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Properties

Label 2-115920-1.1-c1-0-129
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 + 3·11-s + 6·17-s + 19-s − 23-s + 25-s + 2·29-s − 4·31-s − 35-s + 2·37-s − 5·41-s + 6·43-s − 47-s + 49-s − 13·53-s + 3·55-s + 3·59-s − 7·61-s + 2·67-s − 8·71-s + 10·73-s − 3·77-s + 14·79-s + 4·83-s + 6·85-s + 95-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 0.904·11-s + 1.45·17-s + 0.229·19-s − 0.208·23-s + 1/5·25-s + 0.371·29-s − 0.718·31-s − 0.169·35-s + 0.328·37-s − 0.780·41-s + 0.914·43-s − 0.145·47-s + 1/7·49-s − 1.78·53-s + 0.404·55-s + 0.390·59-s − 0.896·61-s + 0.244·67-s − 0.949·71-s + 1.17·73-s − 0.341·77-s + 1.57·79-s + 0.439·83-s + 0.650·85-s + 0.102·95-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: \(1\)
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 \( 1 - 3 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 13 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−13.88314103807805, −13.43550052360503, −12.82090434228335, −12.28372846408121, −12.10229761523280, −11.46161891141412, −10.85077102717860, −10.45941005856962, −9.808024078389277, −9.432566423300768, −9.176104476242096, −8.386702690587769, −7.898841844103238, −7.415688246603544, −6.761952637511084, −6.304349322622645, −5.870794324165064, −5.247045388273584, −4.785438647682976, −3.941555656992180, −3.545127864609594, −2.964706940629624, −2.271963884097935, −1.449183827231648, −1.050170181851970, 0, 1.050170181851970, 1.449183827231648, 2.271963884097935, 2.964706940629624, 3.545127864609594, 3.941555656992180, 4.785438647682976, 5.247045388273584, 5.870794324165064, 6.304349322622645, 6.761952637511084, 7.415688246603544, 7.898841844103238, 8.386702690587769, 9.176104476242096, 9.432566423300768, 9.808024078389277, 10.45941005856962, 10.85077102717860, 11.46161891141412, 12.10229761523280, 12.28372846408121, 12.82090434228335, 13.43550052360503, 13.88314103807805

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