Properties

Label 2-115920-1.1-c1-0-38
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 − 6·13-s − 6·19-s − 23-s + 25-s − 4·29-s + 2·31-s + 35-s − 2·37-s − 2·41-s + 10·43-s + 6·47-s + 49-s − 6·53-s + 2·55-s + 4·59-s − 6·61-s + 6·65-s − 6·67-s − 2·71-s + 6·73-s + 2·77-s + 8·79-s − 14·83-s + 2·89-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 0.603·11-s − 1.66·13-s − 1.37·19-s − 0.208·23-s + 1/5·25-s − 0.742·29-s + 0.359·31-s + 0.169·35-s − 0.328·37-s − 0.312·41-s + 1.52·43-s + 0.875·47-s + 1/7·49-s − 0.824·53-s + 0.269·55-s + 0.520·59-s − 0.768·61-s + 0.744·65-s − 0.733·67-s − 0.237·71-s + 0.702·73-s + 0.227·77-s + 0.900·79-s − 1.53·83-s + 0.211·89-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: \(1\)
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 \( 1 + 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.94123584540536, −13.22618057974131, −12.74637587565530, −12.47030519288376, −12.03111426538398, −11.45869965769029, −10.75269911385244, −10.59637390219539, −9.845882345728363, −9.594357832183166, −8.850417873188275, −8.492839738827155, −7.762194487066159, −7.408389312910276, −7.037657155721928, −6.254014297233636, −5.875219776303366, −5.119678502927463, −4.673712791831653, −4.159529115594760, −3.546814121405125, −2.758756598354821, −2.396720438679741, −1.739020356431464, −0.5896162895501723, 0, 0.5896162895501723, 1.739020356431464, 2.396720438679741, 2.758756598354821, 3.546814121405125, 4.159529115594760, 4.673712791831653, 5.119678502927463, 5.875219776303366, 6.254014297233636, 7.037657155721928, 7.408389312910276, 7.762194487066159, 8.492839738827155, 8.850417873188275, 9.594357832183166, 9.845882345728363, 10.59637390219539, 10.75269911385244, 11.45869965769029, 12.03111426538398, 12.47030519288376, 12.74637587565530, 13.22618057974131, 13.94123584540536

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