Properties

Label 2-119952-1.1-c1-0-122
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  + 5-s − 6·11-s + 4·13-s − 17-s − 19-s − 3·23-s − 4·25-s + 6·29-s − 3·37-s + 12·41-s − 3·43-s − 10·47-s + 2·53-s − 6·55-s + 13·59-s − 10·61-s + 4·65-s + 3·67-s − 3·71-s − 4·73-s − 16·79-s − 12·83-s − 85-s − 89-s − 95-s + 16·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.80·11-s + 1.10·13-s − 0.242·17-s − 0.229·19-s − 0.625·23-s − 4/5·25-s + 1.11·29-s − 0.493·37-s + 1.87·41-s − 0.457·43-s − 1.45·47-s + 0.274·53-s − 0.809·55-s + 1.69·59-s − 1.28·61-s + 0.496·65-s + 0.366·67-s − 0.356·71-s − 0.468·73-s − 1.80·79-s − 1.31·83-s − 0.108·85-s − 0.105·89-s − 0.102·95-s + 1.62·97-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 \( 1 - T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 13 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−13.69301322506156, −13.28199996325081, −12.97041002276723, −12.54079936084577, −11.76523699846468, −11.41308901976401, −10.77294958047411, −10.42669926306392, −10.00398687436072, −9.525496868666685, −8.817513452193382, −8.189169102591125, −8.173361850841840, −7.370009802092457, −6.855482596539036, −6.158428270030403, −5.732367186918244, −5.426579032934738, −4.518781610438055, −4.300309251140437, −3.327832265327461, −2.928803565684923, −2.204390174078659, −1.735750176686110, −0.8063210321544611, 0, 0.8063210321544611, 1.735750176686110, 2.204390174078659, 2.928803565684923, 3.327832265327461, 4.300309251140437, 4.518781610438055, 5.426579032934738, 5.732367186918244, 6.158428270030403, 6.855482596539036, 7.370009802092457, 8.173361850841840, 8.189169102591125, 8.817513452193382, 9.525496868666685, 10.00398687436072, 10.42669926306392, 10.77294958047411, 11.41308901976401, 11.76523699846468, 12.54079936084577, 12.97041002276723, 13.28199996325081, 13.69301322506156

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