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Properties

Label 2-115920-1.1-c1-0-117
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 + 4·13-s − 4·17-s + 2·19-s − 23-s + 25-s − 10·29-s + 6·31-s + 35-s − 6·37-s + 2·41-s + 4·43-s − 10·47-s + 49-s + 10·53-s − 10·61-s + 4·65-s + 12·67-s + 4·71-s − 10·73-s − 8·79-s + 2·83-s − 4·85-s + 4·91-s + 2·95-s − 16·97-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 1.10·13-s − 0.970·17-s + 0.458·19-s − 0.208·23-s + 1/5·25-s − 1.85·29-s + 1.07·31-s + 0.169·35-s − 0.986·37-s + 0.312·41-s + 0.609·43-s − 1.45·47-s + 1/7·49-s + 1.37·53-s − 1.28·61-s + 0.496·65-s + 1.46·67-s + 0.474·71-s − 1.17·73-s − 0.900·79-s + 0.219·83-s − 0.433·85-s + 0.419·91-s + 0.205·95-s − 1.62·97-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 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−13.67077716173906, −13.41385577789628, −13.06757512967711, −12.39573983009702, −11.85271512493581, −11.34087304271095, −10.95092759684176, −10.57370236740839, −9.856458696631904, −9.466928433739392, −8.906177452725299, −8.441766423127153, −8.050221417698116, −7.259393633247965, −6.937485120816920, −6.214756174832552, −5.831866364278927, −5.306617837766062, −4.670750452966392, −4.093211153669292, −3.558691641380312, −2.900068377262914, −2.137425175092416, −1.655198508529400, −0.9692673807118752, 0, 0.9692673807118752, 1.655198508529400, 2.137425175092416, 2.900068377262914, 3.558691641380312, 4.093211153669292, 4.670750452966392, 5.306617837766062, 5.831866364278927, 6.214756174832552, 6.937485120816920, 7.259393633247965, 8.050221417698116, 8.441766423127153, 8.906177452725299, 9.466928433739392, 9.856458696631904, 10.57370236740839, 10.95092759684176, 11.34087304271095, 11.85271512493581, 12.39573983009702, 13.06757512967711, 13.41385577789628, 13.67077716173906

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