Properties

Label 2-119952-1.1-c1-0-13
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 − 5·11-s + 3·13-s − 17-s − 2·19-s − 8·23-s − 25-s + 6·29-s − 4·31-s + 8·37-s + 6·41-s − 4·43-s + 10·47-s − 9·53-s + 10·55-s − 4·59-s − 4·61-s − 6·65-s + 10·67-s − 5·71-s − 2·73-s + 79-s + 12·83-s + 2·85-s + 9·89-s + 4·95-s − 4·97-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.50·11-s + 0.832·13-s − 0.242·17-s − 0.458·19-s − 1.66·23-s − 1/5·25-s + 1.11·29-s − 0.718·31-s + 1.31·37-s + 0.937·41-s − 0.609·43-s + 1.45·47-s − 1.23·53-s + 1.34·55-s − 0.520·59-s − 0.512·61-s − 0.744·65-s + 1.22·67-s − 0.593·71-s − 0.234·73-s + 0.112·79-s + 1.31·83-s + 0.216·85-s + 0.953·89-s + 0.410·95-s − 0.406·97-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

\(L(1)\) \(\approx\) \(0.6656858343\)
\(L(\frac12)\) \(\approx\) \(0.6656858343\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−13.39835349063017, −13.21785389257570, −12.50237001588072, −12.16579039867225, −11.68142768681768, −11.00190451987110, −10.75691266832521, −10.33411261404140, −9.639919973169003, −9.195784658335509, −8.406246341640504, −8.080271108237026, −7.805464262074718, −7.274234880942782, −6.488719714503655, −6.023471606739318, −5.596126538992108, −4.801344451348814, −4.352027120097845, −3.844143378115369, −3.270693908740595, −2.520813270630898, −2.101097006977378, −1.120481240734607, −0.2676210066337778, 0.2676210066337778, 1.120481240734607, 2.101097006977378, 2.520813270630898, 3.270693908740595, 3.844143378115369, 4.352027120097845, 4.801344451348814, 5.596126538992108, 6.023471606739318, 6.488719714503655, 7.274234880942782, 7.805464262074718, 8.080271108237026, 8.406246341640504, 9.195784658335509, 9.639919973169003, 10.33411261404140, 10.75691266832521, 11.00190451987110, 11.68142768681768, 12.16579039867225, 12.50237001588072, 13.21785389257570, 13.39835349063017

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