Properties

Label 2-115920-1.1-c1-0-56
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 − 6·11-s − 4·13-s − 2·17-s + 23-s + 25-s + 8·29-s − 4·31-s − 35-s − 6·37-s + 10·41-s − 6·43-s − 8·47-s + 49-s + 2·53-s − 6·55-s − 2·61-s − 4·65-s + 10·67-s + 6·71-s − 10·73-s + 6·77-s − 8·79-s − 2·85-s − 8·89-s + 4·91-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 1.80·11-s − 1.10·13-s − 0.485·17-s + 0.208·23-s + 1/5·25-s + 1.48·29-s − 0.718·31-s − 0.169·35-s − 0.986·37-s + 1.56·41-s − 0.914·43-s − 1.16·47-s + 1/7·49-s + 0.274·53-s − 0.809·55-s − 0.256·61-s − 0.496·65-s + 1.22·67-s + 0.712·71-s − 1.17·73-s + 0.683·77-s − 0.900·79-s − 0.216·85-s − 0.847·89-s + 0.419·91-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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
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 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−13.78091346290211, −13.25091306577153, −12.91843801379103, −12.52332007822077, −12.02627291679839, −11.36589768872255, −10.79115029720940, −10.43828564432637, −9.853402531954138, −9.706452904806888, −8.864726974008733, −8.472281678610445, −7.860368702018916, −7.404310645492762, −6.854974140992329, −6.406589755618664, −5.605880206247879, −5.301301403887047, −4.746369467948925, −4.280978068156921, −3.247544060036954, −2.892594732321199, −2.315081653525397, −1.793152300511194, −0.6833696944514281, 0, 0.6833696944514281, 1.793152300511194, 2.315081653525397, 2.892594732321199, 3.247544060036954, 4.280978068156921, 4.746369467948925, 5.301301403887047, 5.605880206247879, 6.406589755618664, 6.854974140992329, 7.404310645492762, 7.860368702018916, 8.472281678610445, 8.864726974008733, 9.706452904806888, 9.853402531954138, 10.43828564432637, 10.79115029720940, 11.36589768872255, 12.02627291679839, 12.52332007822077, 12.91843801379103, 13.25091306577153, 13.78091346290211

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