Properties

Label 2-119952-1.1-c1-0-108
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  − 3·5-s + 4·11-s − 13-s + 17-s − 4·19-s − 6·23-s + 4·25-s + 3·29-s + 5·31-s + 8·37-s + 3·41-s − 8·43-s + 47-s − 12·55-s + 11·59-s + 14·61-s + 3·65-s + 2·67-s − 10·71-s − 4·73-s − 4·79-s − 11·83-s − 3·85-s − 4·89-s + 12·95-s + 101-s + 103-s + ⋯
L(s)  = 1  − 1.34·5-s + 1.20·11-s − 0.277·13-s + 0.242·17-s − 0.917·19-s − 1.25·23-s + 4/5·25-s + 0.557·29-s + 0.898·31-s + 1.31·37-s + 0.468·41-s − 1.21·43-s + 0.145·47-s − 1.61·55-s + 1.43·59-s + 1.79·61-s + 0.372·65-s + 0.244·67-s − 1.18·71-s − 0.468·73-s − 0.450·79-s − 1.20·83-s − 0.325·85-s − 0.423·89-s + 1.23·95-s + 0.0995·101-s + 0.0985·103-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: \(1\)
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 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−13.81392810464485, −13.34602116535062, −12.62879048257856, −12.34449810908372, −11.76786990233208, −11.47761337196051, −11.19867363815957, −10.21240343921702, −10.07842813804712, −9.435777969088286, −8.733294856984598, −8.284663477077121, −8.078682257546353, −7.325408547791092, −6.905403336547974, −6.363721578583583, −5.897699973910121, −5.122165520549632, −4.362295575614130, −4.125211539664890, −3.732338867366812, −2.924712616359917, −2.343867361765340, −1.482296668648512, −0.7703961459600005, 0, 0.7703961459600005, 1.482296668648512, 2.343867361765340, 2.924712616359917, 3.732338867366812, 4.125211539664890, 4.362295575614130, 5.122165520549632, 5.897699973910121, 6.363721578583583, 6.905403336547974, 7.325408547791092, 8.078682257546353, 8.284663477077121, 8.733294856984598, 9.435777969088286, 10.07842813804712, 10.21240343921702, 11.19867363815957, 11.47761337196051, 11.76786990233208, 12.34449810908372, 12.62879048257856, 13.34602116535062, 13.81392810464485

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