Properties

Label 2-119952-1.1-c1-0-106
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 − 11-s − 5·13-s + 17-s + 8·19-s − 6·23-s − 4·25-s + 4·29-s − 2·31-s + 3·37-s + 6·41-s − 13·43-s − 2·47-s − 5·53-s + 55-s + 4·59-s + 2·61-s + 5·65-s + 5·67-s + 14·71-s + 11·73-s − 79-s + 15·83-s − 85-s − 5·89-s − 8·95-s − 97-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.301·11-s − 1.38·13-s + 0.242·17-s + 1.83·19-s − 1.25·23-s − 4/5·25-s + 0.742·29-s − 0.359·31-s + 0.493·37-s + 0.937·41-s − 1.98·43-s − 0.291·47-s − 0.686·53-s + 0.134·55-s + 0.520·59-s + 0.256·61-s + 0.620·65-s + 0.610·67-s + 1.66·71-s + 1.28·73-s − 0.112·79-s + 1.64·83-s − 0.108·85-s − 0.529·89-s − 0.820·95-s − 0.101·97-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 \( 1 + T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 13 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−13.93256607221367, −13.38754310389129, −12.68959698175236, −12.35014874617273, −11.77115753264652, −11.61100157683020, −11.01033389596214, −10.16546864662823, −9.970701802551994, −9.523226501595985, −9.044386618140584, −8.012425367102430, −7.952025683791998, −7.592271936524474, −6.795580025332651, −6.461842704087489, −5.599685920002761, −5.161412768449826, −4.836827014928980, −3.939318622517445, −3.613484407522237, −2.839665990391381, −2.351586666068080, −1.613074231374149, −0.7419332396553261, 0, 0.7419332396553261, 1.613074231374149, 2.351586666068080, 2.839665990391381, 3.613484407522237, 3.939318622517445, 4.836827014928980, 5.161412768449826, 5.599685920002761, 6.461842704087489, 6.795580025332651, 7.592271936524474, 7.952025683791998, 8.012425367102430, 9.044386618140584, 9.523226501595985, 9.970701802551994, 10.16546864662823, 11.01033389596214, 11.61100157683020, 11.77115753264652, 12.35014874617273, 12.68959698175236, 13.38754310389129, 13.93256607221367

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