Properties

Label 2-119952-1.1-c1-0-10
Degree $2$
Conductor $119952$
Sign $1$
Analytic cond. $957.821$
Root an. cond. $30.9486$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 2·13-s + 17-s − 25-s − 2·29-s + 8·31-s − 6·37-s − 6·41-s − 4·43-s − 14·53-s + 8·59-s − 14·61-s + 4·65-s − 4·67-s + 8·71-s − 10·73-s − 8·79-s + 16·83-s − 2·85-s + 2·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.554·13-s + 0.242·17-s − 1/5·25-s − 0.371·29-s + 1.43·31-s − 0.986·37-s − 0.937·41-s − 0.609·43-s − 1.92·53-s + 1.04·59-s − 1.79·61-s + 0.496·65-s − 0.488·67-s + 0.949·71-s − 1.17·73-s − 0.900·79-s + 1.75·83-s − 0.216·85-s + 0.211·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(119952\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(957.821\)
Root analytic conductor: \(30.9486\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{119952} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 119952,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5745529514\)
\(L(\frac12)\) \(\approx\) \(0.5745529514\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
17 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−13.54699651980936, −13.12326283227918, −12.43694081429278, −12.05998801978936, −11.74185098958261, −11.26460887138689, −10.61771733040701, −10.20563929760815, −9.703099247909614, −9.159949685355082, −8.540599763091441, −8.136372769014131, −7.582591179979043, −7.299895013238835, −6.460762007314984, −6.261886562035607, −5.320581104246684, −4.925015391504844, −4.403090982212319, −3.747896478879083, −3.266935294852462, −2.695629258439918, −1.882127817260399, −1.236266447670572, −0.2383624658098372, 0.2383624658098372, 1.236266447670572, 1.882127817260399, 2.695629258439918, 3.266935294852462, 3.747896478879083, 4.403090982212319, 4.925015391504844, 5.320581104246684, 6.261886562035607, 6.460762007314984, 7.299895013238835, 7.582591179979043, 8.136372769014131, 8.540599763091441, 9.159949685355082, 9.703099247909614, 10.20563929760815, 10.61771733040701, 11.26460887138689, 11.74185098958261, 12.05998801978936, 12.43694081429278, 13.12326283227918, 13.54699651980936

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