Properties

Label 2-115920-1.1-c1-0-61
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 − 6·13-s − 2·17-s + 4·19-s + 23-s + 25-s + 2·29-s − 4·31-s + 35-s + 2·37-s + 6·41-s − 4·43-s − 8·47-s + 49-s − 10·53-s − 2·61-s + 6·65-s − 4·67-s + 6·73-s − 4·79-s + 12·83-s + 2·85-s − 10·89-s + 6·91-s − 4·95-s + 18·97-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 1.66·13-s − 0.485·17-s + 0.917·19-s + 0.208·23-s + 1/5·25-s + 0.371·29-s − 0.718·31-s + 0.169·35-s + 0.328·37-s + 0.937·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s − 1.37·53-s − 0.256·61-s + 0.744·65-s − 0.488·67-s + 0.702·73-s − 0.450·79-s + 1.31·83-s + 0.216·85-s − 1.05·89-s + 0.628·91-s − 0.410·95-s + 1.82·97-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 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−13.89099554627654, −13.27834401364207, −12.79489360088302, −12.44580574101928, −11.83186398102256, −11.61020339303382, −10.84680623743983, −10.56189800224812, −9.670493398409664, −9.587124621385584, −9.115291266342868, −8.264332242710879, −7.925952202924993, −7.328560759071681, −6.962742294675885, −6.441553582718076, −5.724554024202375, −5.167883324842382, −4.651451430207392, −4.238272540291598, −3.313650149285738, −3.044152324125095, −2.312468866383570, −1.655889911252607, −0.6892149568919948, 0, 0.6892149568919948, 1.655889911252607, 2.312468866383570, 3.044152324125095, 3.313650149285738, 4.238272540291598, 4.651451430207392, 5.167883324842382, 5.724554024202375, 6.441553582718076, 6.962742294675885, 7.328560759071681, 7.925952202924993, 8.264332242710879, 9.115291266342868, 9.587124621385584, 9.670493398409664, 10.56189800224812, 10.84680623743983, 11.61020339303382, 11.83186398102256, 12.44580574101928, 12.79489360088302, 13.27834401364207, 13.89099554627654

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