Properties

Label 2-115920-1.1-c1-0-107
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 − 2·11-s + 6·17-s − 4·19-s + 23-s + 25-s − 2·29-s − 2·31-s + 35-s + 2·41-s − 4·43-s − 4·47-s + 49-s − 2·55-s + 6·59-s + 6·61-s − 8·67-s + 12·71-s − 16·73-s − 2·77-s + 14·79-s − 4·83-s + 6·85-s + 6·89-s − 4·95-s + 18·97-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 0.603·11-s + 1.45·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s − 0.371·29-s − 0.359·31-s + 0.169·35-s + 0.312·41-s − 0.609·43-s − 0.583·47-s + 1/7·49-s − 0.269·55-s + 0.781·59-s + 0.768·61-s − 0.977·67-s + 1.42·71-s − 1.87·73-s − 0.227·77-s + 1.57·79-s − 0.439·83-s + 0.650·85-s + 0.635·89-s − 0.410·95-s + 1.82·97-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: \(1\)
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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−13.84456638561633, −13.29025802284135, −12.88764250065695, −12.53689336257536, −11.78065980191058, −11.56716721714362, −10.76855063744355, −10.44837284059158, −10.04768978967778, −9.446967221158470, −8.953532947369846, −8.405065759156169, −7.806068312073224, −7.595985603206286, −6.771082207210420, −6.360621407394934, −5.653234845143625, −5.288565729970645, −4.829288103531927, −4.051458607597956, −3.527536909384185, −2.846213564275542, −2.250062144290115, −1.619208361706768, −0.9408287613334088, 0, 0.9408287613334088, 1.619208361706768, 2.250062144290115, 2.846213564275542, 3.527536909384185, 4.051458607597956, 4.829288103531927, 5.288565729970645, 5.653234845143625, 6.360621407394934, 6.771082207210420, 7.595985603206286, 7.806068312073224, 8.405065759156169, 8.953532947369846, 9.446967221158470, 10.04768978967778, 10.44837284059158, 10.76855063744355, 11.56716721714362, 11.78065980191058, 12.53689336257536, 12.88764250065695, 13.29025802284135, 13.84456638561633

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