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Properties

Label 2-115920-1.1-c1-0-138
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 7-s − 6·13-s + 2·17-s − 4·19-s − 23-s + 25-s − 6·29-s − 4·31-s − 35-s + 2·37-s − 10·41-s − 8·43-s + 4·47-s + 49-s − 10·53-s + 4·59-s + 2·61-s − 6·65-s − 8·67-s − 14·73-s − 8·79-s − 12·83-s + 2·85-s − 14·89-s + 6·91-s − 4·95-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 1.66·13-s + 0.485·17-s − 0.917·19-s − 0.208·23-s + 1/5·25-s − 1.11·29-s − 0.718·31-s − 0.169·35-s + 0.328·37-s − 1.56·41-s − 1.21·43-s + 0.583·47-s + 1/7·49-s − 1.37·53-s + 0.520·59-s + 0.256·61-s − 0.744·65-s − 0.977·67-s − 1.63·73-s − 0.900·79-s − 1.31·83-s + 0.216·85-s − 1.48·89-s + 0.628·91-s − 0.410·95-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: \(2\)
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 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−14.16910258876053, −13.53593318756507, −13.03357564306968, −12.69597899729826, −12.24338496046258, −11.66929799354593, −11.26982956286187, −10.49798435942728, −10.13127558624570, −9.781993740315131, −9.260499595947050, −8.760881564903881, −8.146487545643118, −7.605425637151886, −7.038919165289941, −6.728476449790900, −5.993792638063812, −5.477636191340080, −5.073095111092894, −4.376852075604450, −3.873053942606258, −3.041328067584140, −2.672340706177566, −1.866012540308688, −1.457964254778044, 0, 0, 1.457964254778044, 1.866012540308688, 2.672340706177566, 3.041328067584140, 3.873053942606258, 4.376852075604450, 5.073095111092894, 5.477636191340080, 5.993792638063812, 6.728476449790900, 7.038919165289941, 7.605425637151886, 8.146487545643118, 8.760881564903881, 9.260499595947050, 9.781993740315131, 10.13127558624570, 10.49798435942728, 11.26982956286187, 11.66929799354593, 12.24338496046258, 12.69597899729826, 13.03357564306968, 13.53593318756507, 14.16910258876053

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