Properties

Label 2-119952-1.1-c1-0-132
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 − 13-s − 17-s + 8·23-s − 4·25-s − 5·29-s + 3·31-s + 10·37-s + 7·41-s − 10·43-s + 5·47-s + 6·53-s − 11·59-s + 8·61-s + 65-s − 2·67-s + 4·71-s − 4·73-s − 9·83-s + 85-s − 2·89-s + 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.277·13-s − 0.242·17-s + 1.66·23-s − 4/5·25-s − 0.928·29-s + 0.538·31-s + 1.64·37-s + 1.09·41-s − 1.52·43-s + 0.729·47-s + 0.824·53-s − 1.43·59-s + 1.02·61-s + 0.124·65-s − 0.244·67-s + 0.474·71-s − 0.468·73-s − 0.987·83-s + 0.108·85-s − 0.211·89-s + 1.21·97-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 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 5 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−13.70722323439393, −13.15707046455744, −13.05214308423093, −12.28800191426718, −11.87939448582365, −11.32392581183631, −11.03657531781155, −10.49163770588406, −9.813924921579422, −9.421962349517334, −8.972285794980308, −8.342581048521788, −7.869795398767299, −7.373500337261729, −6.933307415474701, −6.337331699396893, −5.728098444108324, −5.237823361741292, −4.569920129638959, −4.146313132273053, −3.536832834471837, −2.830946777507197, −2.402555479785818, −1.514221655210548, −0.8322836064340910, 0, 0.8322836064340910, 1.514221655210548, 2.402555479785818, 2.830946777507197, 3.536832834471837, 4.146313132273053, 4.569920129638959, 5.237823361741292, 5.728098444108324, 6.337331699396893, 6.933307415474701, 7.373500337261729, 7.869795398767299, 8.342581048521788, 8.972285794980308, 9.421962349517334, 9.813924921579422, 10.49163770588406, 11.03657531781155, 11.32392581183631, 11.87939448582365, 12.28800191426718, 13.05214308423093, 13.15707046455744, 13.70722323439393

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