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Properties

Label 2-115920-1.1-c1-0-93
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 + 2·13-s + 6·17-s − 4·19-s + 23-s + 25-s − 10·29-s + 8·31-s + 35-s − 2·37-s + 6·41-s − 8·47-s + 49-s + 10·53-s + 12·59-s − 2·61-s − 2·65-s − 8·67-s − 12·71-s − 6·73-s + 8·79-s + 12·83-s − 6·85-s − 6·89-s − 2·91-s + 4·95-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s + 0.554·13-s + 1.45·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s − 1.85·29-s + 1.43·31-s + 0.169·35-s − 0.328·37-s + 0.937·41-s − 1.16·47-s + 1/7·49-s + 1.37·53-s + 1.56·59-s − 0.256·61-s − 0.248·65-s − 0.977·67-s − 1.42·71-s − 0.702·73-s + 0.900·79-s + 1.31·83-s − 0.650·85-s − 0.635·89-s − 0.209·91-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: \(1\)
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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−13.59417834749203, −13.44972605208898, −12.87251885625959, −12.37969269620414, −11.91527132839241, −11.48180526332425, −10.93427041483265, −10.41629398329988, −9.995585658500361, −9.470015488323885, −8.859492242829780, −8.461073370836545, −7.849513506610161, −7.477037868448399, −6.877240861546239, −6.299367729625796, −5.791846470486037, −5.334799714806280, −4.603910899086011, −3.980348012592994, −3.590888250754062, −2.980120992624927, −2.332631145343692, −1.501745508692475, −0.8517631551451006, 0, 0.8517631551451006, 1.501745508692475, 2.332631145343692, 2.980120992624927, 3.590888250754062, 3.980348012592994, 4.603910899086011, 5.334799714806280, 5.791846470486037, 6.299367729625796, 6.877240861546239, 7.477037868448399, 7.849513506610161, 8.461073370836545, 8.859492242829780, 9.470015488323885, 9.995585658500361, 10.41629398329988, 10.93427041483265, 11.48180526332425, 11.91527132839241, 12.37969269620414, 12.87251885625959, 13.44972605208898, 13.59417834749203

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