Properties

Label 2-119952-1.1-c1-0-105
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 − 2·11-s + 4·13-s − 17-s + 2·19-s − 6·23-s − 25-s + 4·31-s + 2·37-s + 6·41-s − 12·43-s − 12·47-s − 10·53-s + 4·55-s + 4·59-s + 2·61-s − 8·65-s − 12·67-s − 2·71-s + 14·73-s + 8·79-s + 4·83-s + 2·85-s + 18·89-s − 4·95-s + 10·97-s + 101-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.603·11-s + 1.10·13-s − 0.242·17-s + 0.458·19-s − 1.25·23-s − 1/5·25-s + 0.718·31-s + 0.328·37-s + 0.937·41-s − 1.82·43-s − 1.75·47-s − 1.37·53-s + 0.539·55-s + 0.520·59-s + 0.256·61-s − 0.992·65-s − 1.46·67-s − 0.237·71-s + 1.63·73-s + 0.900·79-s + 0.439·83-s + 0.216·85-s + 1.90·89-s − 0.410·95-s + 1.01·97-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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−13.62762633819338, −13.35167570213284, −12.96451709196248, −12.21990919378944, −11.79563665139810, −11.52400370011956, −10.89114674156579, −10.55966851199697, −9.805938752471475, −9.580841447122850, −8.733709008475798, −8.300420557367274, −7.896511236463932, −7.597431384411136, −6.815082617191856, −6.125009743848193, −6.068788061107420, −4.911282050458335, −4.851899656835780, −3.938697278450580, −3.557359975199127, −3.077442279798105, −2.210519536661692, −1.610283256200408, −0.7343794416947569, 0, 0.7343794416947569, 1.610283256200408, 2.210519536661692, 3.077442279798105, 3.557359975199127, 3.938697278450580, 4.851899656835780, 4.911282050458335, 6.068788061107420, 6.125009743848193, 6.815082617191856, 7.597431384411136, 7.896511236463932, 8.300420557367274, 8.733709008475798, 9.580841447122850, 9.805938752471475, 10.55966851199697, 10.89114674156579, 11.52400370011956, 11.79563665139810, 12.21990919378944, 12.96451709196248, 13.35167570213284, 13.62762633819338

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