Properties

Label 2-119952-1.1-c1-0-140
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  + 4·11-s + 5·13-s + 17-s − 7·19-s − 6·23-s − 5·25-s − 6·29-s − 31-s + 5·37-s + 6·41-s + 43-s − 2·47-s − 6·53-s − 4·59-s − 10·61-s − 13·67-s + 14·71-s + 11·73-s + 5·79-s − 2·83-s + 8·89-s + 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.20·11-s + 1.38·13-s + 0.242·17-s − 1.60·19-s − 1.25·23-s − 25-s − 1.11·29-s − 0.179·31-s + 0.821·37-s + 0.937·41-s + 0.152·43-s − 0.291·47-s − 0.824·53-s − 0.520·59-s − 1.28·61-s − 1.58·67-s + 1.66·71-s + 1.28·73-s + 0.562·79-s − 0.219·83-s + 0.847·89-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−13.80799005873139, −13.35543280986683, −12.83754807648368, −12.37759775720926, −11.85478661398389, −11.36168004239852, −10.91739281058238, −10.56071265037680, −9.837366008669920, −9.292928022899140, −9.040790080275564, −8.397473081405689, −7.830429185188512, −7.585367823473967, −6.546632671513722, −6.255864002017306, −6.049103839760037, −5.308950207155501, −4.428493243869850, −3.975146059397663, −3.762180664215882, −2.970493665062359, −1.961733677003984, −1.774585775999115, −0.8913375140896693, 0, 0.8913375140896693, 1.774585775999115, 1.961733677003984, 2.970493665062359, 3.762180664215882, 3.975146059397663, 4.428493243869850, 5.308950207155501, 6.049103839760037, 6.255864002017306, 6.546632671513722, 7.585367823473967, 7.830429185188512, 8.397473081405689, 9.040790080275564, 9.292928022899140, 9.837366008669920, 10.56071265037680, 10.91739281058238, 11.36168004239852, 11.85478661398389, 12.37759775720926, 12.83754807648368, 13.35543280986683, 13.80799005873139

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