Properties

Label 2-119952-1.1-c1-0-129
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 − 2·11-s + 3·13-s + 17-s + 2·19-s + 4·23-s − 4·25-s − 3·29-s + 31-s − 4·37-s − 3·41-s + 2·43-s − 47-s − 6·53-s + 2·55-s − 11·59-s − 2·61-s − 3·65-s + 4·67-s + 10·71-s + 2·73-s + 16·79-s + 3·83-s − 85-s − 2·95-s − 2·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.603·11-s + 0.832·13-s + 0.242·17-s + 0.458·19-s + 0.834·23-s − 4/5·25-s − 0.557·29-s + 0.179·31-s − 0.657·37-s − 0.468·41-s + 0.304·43-s − 0.145·47-s − 0.824·53-s + 0.269·55-s − 1.43·59-s − 0.256·61-s − 0.372·65-s + 0.488·67-s + 1.18·71-s + 0.234·73-s + 1.80·79-s + 0.329·83-s − 0.108·85-s − 0.205·95-s − 0.203·97-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: $\chi_{119952} (1, \cdot )$
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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.74089959813885, −13.42396469298490, −12.79113440446213, −12.38758630349385, −11.88783599749098, −11.31869234264717, −10.92780057944799, −10.56453316363828, −9.862357025974813, −9.419239283444778, −8.917548232616131, −8.330236853649469, −7.811015274068497, −7.559820301186684, −6.813513062241591, −6.342165947673749, −5.744970280246068, −5.165639471131433, −4.768935307711408, −3.964306809736865, −3.489285528059849, −3.053363030390378, −2.235935645409269, −1.568361059983523, −0.8256166502174795, 0, 0.8256166502174795, 1.568361059983523, 2.235935645409269, 3.053363030390378, 3.489285528059849, 3.964306809736865, 4.768935307711408, 5.165639471131433, 5.744970280246068, 6.342165947673749, 6.813513062241591, 7.559820301186684, 7.811015274068497, 8.330236853649469, 8.917548232616131, 9.419239283444778, 9.862357025974813, 10.56453316363828, 10.92780057944799, 11.31869234264717, 11.88783599749098, 12.38758630349385, 12.79113440446213, 13.42396469298490, 13.74089959813885

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