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Properties

Label 2-115920-1.1-c1-0-111
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 − 4·17-s − 4·19-s + 23-s + 25-s + 8·29-s + 4·31-s − 35-s + 2·37-s − 4·43-s − 4·47-s + 49-s − 8·59-s − 6·61-s + 6·65-s − 4·67-s + 12·71-s − 4·73-s − 12·79-s + 4·83-s − 4·85-s + 6·89-s − 6·91-s − 4·95-s + 6·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 1.66·13-s − 0.970·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s + 1.48·29-s + 0.718·31-s − 0.169·35-s + 0.328·37-s − 0.609·43-s − 0.583·47-s + 1/7·49-s − 1.04·59-s − 0.768·61-s + 0.744·65-s − 0.488·67-s + 1.42·71-s − 0.468·73-s − 1.35·79-s + 0.439·83-s − 0.433·85-s + 0.635·89-s − 0.628·91-s − 0.410·95-s + 0.609·97-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: 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 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−13.79696754293498, −13.41262703829185, −12.88382279597778, −12.59143421836861, −11.85213621453030, −11.34156032140045, −10.95784527173142, −10.30292382932898, −10.18080504070999, −9.317043682533711, −8.899276555345390, −8.510277899533092, −8.079689848076920, −7.339192187185747, −6.572677060636611, −6.302473362029214, −6.112661233931627, −5.186239201817234, −4.634536326500634, −4.153841148059692, −3.473138723127877, −2.904159507521208, −2.292274798930707, −1.544277107021401, −0.9549446564161390, 0, 0.9549446564161390, 1.544277107021401, 2.292274798930707, 2.904159507521208, 3.473138723127877, 4.153841148059692, 4.634536326500634, 5.186239201817234, 6.112661233931627, 6.302473362029214, 6.572677060636611, 7.339192187185747, 8.079689848076920, 8.510277899533092, 8.899276555345390, 9.317043682533711, 10.18080504070999, 10.30292382932898, 10.95784527173142, 11.34156032140045, 11.85213621453030, 12.59143421836861, 12.88382279597778, 13.41262703829185, 13.79696754293498

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