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Properties

Label 2-115920-1.1-c1-0-113
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 − 4·13-s + 6·17-s − 8·19-s + 23-s + 25-s − 4·31-s + 35-s + 2·37-s + 2·41-s + 10·43-s + 49-s + 2·53-s + 2·55-s − 2·61-s − 4·65-s + 10·67-s + 6·71-s − 2·73-s + 2·77-s − 8·79-s + 16·83-s + 6·85-s − 4·91-s − 8·95-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 0.603·11-s − 1.10·13-s + 1.45·17-s − 1.83·19-s + 0.208·23-s + 1/5·25-s − 0.718·31-s + 0.169·35-s + 0.328·37-s + 0.312·41-s + 1.52·43-s + 1/7·49-s + 0.274·53-s + 0.269·55-s − 0.256·61-s − 0.496·65-s + 1.22·67-s + 0.712·71-s − 0.234·73-s + 0.227·77-s − 0.900·79-s + 1.75·83-s + 0.650·85-s − 0.419·91-s − 0.820·95-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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−13.96309129379423, −13.36794878827394, −12.67615775159561, −12.45347557207365, −12.09959863517010, −11.32253712755604, −10.94905537119782, −10.41298433489091, −9.946730187248877, −9.428188344610019, −9.028784693398210, −8.439780137004280, −7.828102342863911, −7.485030238906985, −6.799814405730424, −6.360890144279625, −5.730622626089103, −5.277921118794922, −4.719566501772408, −4.046286869197270, −3.677693363307731, −2.645968344554332, −2.383747035336971, −1.585014986121534, −0.9594358146963933, 0, 0.9594358146963933, 1.585014986121534, 2.383747035336971, 2.645968344554332, 3.677693363307731, 4.046286869197270, 4.719566501772408, 5.277921118794922, 5.730622626089103, 6.360890144279625, 6.799814405730424, 7.485030238906985, 7.828102342863911, 8.439780137004280, 9.028784693398210, 9.428188344610019, 9.946730187248877, 10.41298433489091, 10.94905537119782, 11.32253712755604, 12.09959863517010, 12.45347557207365, 12.67615775159561, 13.36794878827394, 13.96309129379423

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