Properties

Label 2-119952-1.1-c1-0-113
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  − 4·5-s + 4·11-s + 13-s − 17-s − 19-s + 2·23-s + 11·25-s − 6·29-s + 9·31-s − 11·37-s − 10·41-s + 7·43-s + 6·47-s + 6·53-s − 16·55-s + 8·59-s + 6·61-s − 4·65-s − 3·67-s − 2·71-s − 9·73-s + 3·79-s + 6·83-s + 4·85-s + 4·95-s + 2·97-s + 101-s + ⋯
L(s)  = 1  − 1.78·5-s + 1.20·11-s + 0.277·13-s − 0.242·17-s − 0.229·19-s + 0.417·23-s + 11/5·25-s − 1.11·29-s + 1.61·31-s − 1.80·37-s − 1.56·41-s + 1.06·43-s + 0.875·47-s + 0.824·53-s − 2.15·55-s + 1.04·59-s + 0.768·61-s − 0.496·65-s − 0.366·67-s − 0.237·71-s − 1.05·73-s + 0.337·79-s + 0.658·83-s + 0.433·85-s + 0.410·95-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: $\chi_{119952} (1, \cdot )$
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 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−13.70886151666889, −13.41420412509155, −12.69580924508277, −12.12668046677650, −11.90630850133137, −11.54390060185931, −10.98818844667415, −10.54576559869678, −9.999957269820428, −9.222065058818169, −8.742940753725498, −8.501565480418434, −7.927890812133323, −7.253284660595924, −6.967194992177944, −6.520876240002346, −5.770835972652806, −5.082675019499646, −4.523844287660777, −3.917719169282226, −3.723983076546228, −3.096213543197652, −2.313289068049964, −1.411717681090370, −0.7846890862291297, 0, 0.7846890862291297, 1.411717681090370, 2.313289068049964, 3.096213543197652, 3.723983076546228, 3.917719169282226, 4.523844287660777, 5.082675019499646, 5.770835972652806, 6.520876240002346, 6.967194992177944, 7.253284660595924, 7.927890812133323, 8.501565480418434, 8.742940753725498, 9.222065058818169, 9.999957269820428, 10.54576559869678, 10.98818844667415, 11.54390060185931, 11.90630850133137, 12.12668046677650, 12.69580924508277, 13.41420412509155, 13.70886151666889

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