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Properties

Label 2-119952-1.1-c1-0-139
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 4·11-s + 2·13-s + 17-s + 4·19-s + 6·23-s − 25-s + 10·29-s + 8·31-s + 6·41-s + 8·43-s − 4·47-s − 14·53-s + 8·55-s + 14·59-s − 14·61-s − 4·65-s − 4·67-s − 10·71-s − 14·73-s − 4·79-s + 6·83-s − 2·85-s + 2·89-s − 8·95-s − 2·97-s + 101-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.20·11-s + 0.554·13-s + 0.242·17-s + 0.917·19-s + 1.25·23-s − 1/5·25-s + 1.85·29-s + 1.43·31-s + 0.937·41-s + 1.21·43-s − 0.583·47-s − 1.92·53-s + 1.07·55-s + 1.82·59-s − 1.79·61-s − 0.496·65-s − 0.488·67-s − 1.18·71-s − 1.63·73-s − 0.450·79-s + 0.658·83-s − 0.216·85-s + 0.211·89-s − 0.820·95-s − 0.203·97-s + 0.0995·101-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 119952,\ (\ :1/2),\ -1)\)

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

−13.69795045500263, −13.35037693593063, −12.88025032531610, −12.16708460739345, −12.03635695340389, −11.33377052140336, −10.96206942716796, −10.46827291443566, −9.984976811385138, −9.438405824611944, −8.791266576209006, −8.310101490993942, −7.881913194225604, −7.475964523756665, −6.979606403076183, −6.226637209338121, −5.843441094661044, −5.081629444845265, −4.636835630051220, −4.227582070367137, −3.322268092162750, −2.949067745862572, −2.535995344237527, −1.344512665994370, −0.8830700964014068, 0, 0.8830700964014068, 1.344512665994370, 2.535995344237527, 2.949067745862572, 3.322268092162750, 4.227582070367137, 4.636835630051220, 5.081629444845265, 5.843441094661044, 6.226637209338121, 6.979606403076183, 7.475964523756665, 7.881913194225604, 8.310101490993942, 8.791266576209006, 9.438405824611944, 9.984976811385138, 10.46827291443566, 10.96206942716796, 11.33377052140336, 12.03635695340389, 12.16708460739345, 12.88025032531610, 13.35037693593063, 13.69795045500263

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