Properties

Label 2-115920-1.1-c1-0-112
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 + 4·11-s − 6·17-s − 23-s + 25-s + 8·29-s + 8·31-s − 35-s − 2·37-s − 2·41-s − 8·43-s + 49-s + 4·55-s + 10·59-s + 12·67-s + 4·71-s − 4·77-s − 10·79-s − 2·83-s − 6·85-s − 10·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 1.20·11-s − 1.45·17-s − 0.208·23-s + 1/5·25-s + 1.48·29-s + 1.43·31-s − 0.169·35-s − 0.328·37-s − 0.312·41-s − 1.21·43-s + 1/7·49-s + 0.539·55-s + 1.30·59-s + 1.46·67-s + 0.474·71-s − 0.455·77-s − 1.12·79-s − 0.219·83-s − 0.650·85-s − 1.05·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−13.80308188736945, −13.50304558037071, −12.85515851299196, −12.44343502588582, −11.81964707036894, −11.52076172861525, −10.99497121696884, −10.28085827272409, −9.935965319640171, −9.543339150032858, −8.829576116680592, −8.530139085735598, −8.130510209105101, −7.115498589207272, −6.730523051573487, −6.468879931289247, −5.957549374183904, −5.140293932957320, −4.691584316120916, −4.078803104848793, −3.608868804424151, −2.754763952544826, −2.371622001808814, −1.533894811534820, −0.9581778959813201, 0, 0.9581778959813201, 1.533894811534820, 2.371622001808814, 2.754763952544826, 3.608868804424151, 4.078803104848793, 4.691584316120916, 5.140293932957320, 5.957549374183904, 6.468879931289247, 6.730523051573487, 7.115498589207272, 8.130510209105101, 8.530139085735598, 8.829576116680592, 9.543339150032858, 9.935965319640171, 10.28085827272409, 10.99497121696884, 11.52076172861525, 11.81964707036894, 12.44343502588582, 12.85515851299196, 13.50304558037071, 13.80308188736945

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