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Properties

Label 2-115920-1.1-c1-0-60
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 − 11-s + 4·13-s − 6·17-s − 7·19-s − 23-s + 25-s + 6·29-s − 4·31-s + 35-s − 2·37-s − 9·41-s − 2·43-s + 7·47-s + 49-s − 5·53-s + 55-s + 7·59-s − 7·61-s − 4·65-s + 2·67-s + 4·71-s − 10·73-s + 77-s + 6·79-s + 16·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 0.301·11-s + 1.10·13-s − 1.45·17-s − 1.60·19-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 − 1.40·41-s − 0.304·43-s + 1.02·47-s + 1/7·49-s − 0.686·53-s + 0.134·55-s + 0.911·59-s − 0.896·61-s − 0.496·65-s + 0.244·67-s + 0.474·71-s − 1.17·73-s + 0.113·77-s + 0.675·79-s + 1.75·83-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 \( 1 + T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.64636197573836, −13.38139085389670, −12.95389383434358, −12.37537187013602, −11.99908163535025, −11.29653623659908, −10.91544083874523, −10.53071697204225, −10.11523030950007, −9.270514853004609, −8.867785618251875, −8.410183118544833, −8.133040370855080, −7.303463424916536, −6.716104963889051, −6.457720423103407, −5.917103782693349, −5.187795719684261, −4.529010756709068, −4.152235046383521, −3.546701593837654, −2.978142561452966, −2.182569090573861, −1.740926193790851, −0.6881394299182240, 0, 0.6881394299182240, 1.740926193790851, 2.182569090573861, 2.978142561452966, 3.546701593837654, 4.152235046383521, 4.529010756709068, 5.187795719684261, 5.917103782693349, 6.457720423103407, 6.716104963889051, 7.303463424916536, 8.133040370855080, 8.410183118544833, 8.867785618251875, 9.270514853004609, 10.11523030950007, 10.53071697204225, 10.91544083874523, 11.29653623659908, 11.99908163535025, 12.37537187013602, 12.95389383434358, 13.38139085389670, 13.64636197573836

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