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Properties

Label 2-119952-1.1-c1-0-124
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 + 3·11-s − 6·13-s + 17-s − 6·19-s − 2·23-s − 4·25-s − 3·29-s + 7·31-s + 4·37-s + 6·41-s − 6·43-s − 6·47-s − 5·53-s + 3·55-s − 59-s − 6·65-s + 12·67-s − 6·71-s − 6·73-s + 5·79-s + 11·83-s + 85-s + 14·89-s − 6·95-s + 7·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.904·11-s − 1.66·13-s + 0.242·17-s − 1.37·19-s − 0.417·23-s − 4/5·25-s − 0.557·29-s + 1.25·31-s + 0.657·37-s + 0.937·41-s − 0.914·43-s − 0.875·47-s − 0.686·53-s + 0.404·55-s − 0.130·59-s − 0.744·65-s + 1.46·67-s − 0.712·71-s − 0.702·73-s + 0.562·79-s + 1.20·83-s + 0.108·85-s + 1.48·89-s − 0.615·95-s + 0.710·97-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 \( 1 - T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−13.81985779247484, −13.27999523232549, −12.89256948296321, −12.22172493313000, −12.02114043120413, −11.45279332922052, −10.91935269816810, −10.26788195879021, −9.842216568741144, −9.537173483447100, −9.040046061075669, −8.325008974037028, −7.922034808737141, −7.376525744251058, −6.768894750910640, −6.231666214836052, −5.978161268843428, −5.099249936196090, −4.683290026225761, −4.167963571734430, −3.538759924488923, −2.792441384709675, −2.128869052248872, −1.819755839426664, −0.8107747288206403, 0, 0.8107747288206403, 1.819755839426664, 2.128869052248872, 2.792441384709675, 3.538759924488923, 4.167963571734430, 4.683290026225761, 5.099249936196090, 5.978161268843428, 6.231666214836052, 6.768894750910640, 7.376525744251058, 7.922034808737141, 8.325008974037028, 9.040046061075669, 9.537173483447100, 9.842216568741144, 10.26788195879021, 10.91935269816810, 11.45279332922052, 12.02114043120413, 12.22172493313000, 12.89256948296321, 13.27999523232549, 13.81985779247484

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