Properties

Label 2-115920-1.1-c1-0-22
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 7-s − 4·11-s + 2·13-s + 2·17-s + 4·19-s − 23-s + 25-s − 2·29-s + 35-s + 6·37-s + 6·41-s + 8·43-s + 4·47-s + 49-s − 2·53-s + 4·55-s + 12·59-s − 2·61-s − 2·65-s − 8·67-s + 8·71-s − 2·73-s + 4·77-s − 8·79-s + 8·83-s − 2·85-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s − 0.371·29-s + 0.169·35-s + 0.986·37-s + 0.937·41-s + 1.21·43-s + 0.583·47-s + 1/7·49-s − 0.274·53-s + 0.539·55-s + 1.56·59-s − 0.256·61-s − 0.248·65-s − 0.977·67-s + 0.949·71-s − 0.234·73-s + 0.455·77-s − 0.900·79-s + 0.878·83-s − 0.216·85-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: \(0\)
Selberg data: \((2,\ 115920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.125124818\)
\(L(\frac12)\) \(\approx\) \(2.125124818\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−13.59139301676950, −12.99606341235292, −12.76380020963870, −12.17102204827296, −11.65364048193276, −11.15489007532000, −10.72429386272653, −10.20771339421906, −9.725067048697681, −9.194056553344360, −8.697666614980045, −8.022321865245024, −7.618523599269354, −7.362094256484874, −6.580609964359352, −5.929926708654718, −5.590622848634291, −5.016479725689010, −4.306977328215248, −3.819814230630272, −3.157536259437812, −2.701050331080393, −2.041777618622820, −1.030684326688249, −0.5260437575680942, 0.5260437575680942, 1.030684326688249, 2.041777618622820, 2.701050331080393, 3.157536259437812, 3.819814230630272, 4.306977328215248, 5.016479725689010, 5.590622848634291, 5.929926708654718, 6.580609964359352, 7.362094256484874, 7.618523599269354, 8.022321865245024, 8.697666614980045, 9.194056553344360, 9.725067048697681, 10.20771339421906, 10.72429386272653, 11.15489007532000, 11.65364048193276, 12.17102204827296, 12.76380020963870, 12.99606341235292, 13.59139301676950

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