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Properties

Label 2-115920-1.1-c1-0-99
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 − 2·11-s + 2·13-s + 2·17-s + 4·19-s + 23-s + 25-s + 8·29-s − 2·31-s + 35-s + 4·37-s + 2·41-s + 8·43-s − 12·47-s + 49-s + 10·53-s + 2·55-s + 4·59-s − 8·61-s − 2·65-s + 8·67-s − 4·71-s − 6·73-s + 2·77-s − 8·79-s − 6·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 0.603·11-s + 0.554·13-s + 0.485·17-s + 0.917·19-s + 0.208·23-s + 1/5·25-s + 1.48·29-s − 0.359·31-s + 0.169·35-s + 0.657·37-s + 0.312·41-s + 1.21·43-s − 1.75·47-s + 1/7·49-s + 1.37·53-s + 0.269·55-s + 0.520·59-s − 1.02·61-s − 0.248·65-s + 0.977·67-s − 0.474·71-s − 0.702·73-s + 0.227·77-s − 0.900·79-s − 0.658·83-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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
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

−13.81509954789284, −13.26862309794639, −12.94070807302286, −12.40264879335332, −11.84990912985944, −11.52130536299633, −10.92009364442374, −10.41071792311466, −10.01984455501495, −9.434564275478665, −8.945057726950971, −8.353326448866665, −7.903753831519687, −7.461196741228117, −6.872559914246640, −6.365034672900860, −5.726576727524082, −5.296859466232260, −4.657581038818939, −4.064961030811803, −3.494271578594728, −2.868766461908293, −2.514893795537129, −1.389304709416970, −0.9062515079423203, 0, 0.9062515079423203, 1.389304709416970, 2.514893795537129, 2.868766461908293, 3.494271578594728, 4.064961030811803, 4.657581038818939, 5.296859466232260, 5.726576727524082, 6.365034672900860, 6.872559914246640, 7.461196741228117, 7.903753831519687, 8.353326448866665, 8.945057726950971, 9.434564275478665, 10.01984455501495, 10.41071792311466, 10.92009364442374, 11.52130536299633, 11.84990912985944, 12.40264879335332, 12.94070807302286, 13.26862309794639, 13.81509954789284

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