Properties

Label 2-115920-1.1-c1-0-132
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 + 4·13-s + 4·19-s + 23-s + 25-s − 2·29-s − 4·31-s + 35-s − 2·37-s − 12·41-s − 4·43-s − 8·47-s + 49-s − 10·53-s + 4·55-s + 6·59-s − 12·61-s + 4·65-s − 4·67-s − 12·71-s + 10·73-s + 4·77-s − 2·79-s + 14·83-s + 6·89-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 1.20·11-s + 1.10·13-s + 0.917·19-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 − 1.87·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s − 1.37·53-s + 0.539·55-s + 0.781·59-s − 1.53·61-s + 0.496·65-s − 0.488·67-s − 1.42·71-s + 1.17·73-s + 0.455·77-s − 0.225·79-s + 1.53·83-s + 0.635·89-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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.87433022901236, −13.37590904618917, −13.03662961259033, −12.31623091730781, −11.81007974544239, −11.47345173252551, −11.01310711643562, −10.42863608734072, −9.946582328125381, −9.313604269069554, −9.013021042040114, −8.528444526000400, −7.916810405552838, −7.410533938712576, −6.662744984664297, −6.438087667802850, −5.833721007840751, −5.174824537280993, −4.810413266276228, −4.016134257930991, −3.407337419806602, −3.169804466442259, −2.027355324234249, −1.526830096355585, −1.139986498242309, 0, 1.139986498242309, 1.526830096355585, 2.027355324234249, 3.169804466442259, 3.407337419806602, 4.016134257930991, 4.810413266276228, 5.174824537280993, 5.833721007840751, 6.438087667802850, 6.662744984664297, 7.410533938712576, 7.916810405552838, 8.528444526000400, 9.013021042040114, 9.313604269069554, 9.946582328125381, 10.42863608734072, 11.01310711643562, 11.47345173252551, 11.81007974544239, 12.31623091730781, 13.03662961259033, 13.37590904618917, 13.87433022901236

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