Properties

Label 2-119952-1.1-c1-0-123
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 − 5·11-s + 5·13-s + 17-s − 4·19-s − 6·23-s − 4·25-s − 2·31-s + 37-s + 2·41-s − 43-s + 6·47-s + 9·53-s − 5·55-s − 4·59-s − 6·61-s + 5·65-s + 67-s − 6·71-s + 7·73-s + 79-s − 83-s + 85-s + 7·89-s − 4·95-s + 3·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.50·11-s + 1.38·13-s + 0.242·17-s − 0.917·19-s − 1.25·23-s − 4/5·25-s − 0.359·31-s + 0.164·37-s + 0.312·41-s − 0.152·43-s + 0.875·47-s + 1.23·53-s − 0.674·55-s − 0.520·59-s − 0.768·61-s + 0.620·65-s + 0.122·67-s − 0.712·71-s + 0.819·73-s + 0.112·79-s − 0.109·83-s + 0.108·85-s + 0.741·89-s − 0.410·95-s + 0.304·97-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 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−13.54810655100813, −13.48634255427193, −12.95508914753298, −12.40711149232765, −11.93718684392254, −11.29199528519012, −10.81138832099297, −10.38129071095207, −10.10739124478802, −9.416133815982729, −8.840453651728759, −8.423260656857132, −7.803459344228786, −7.610532397345585, −6.743863671908655, −6.135241241267493, −5.818713512225328, −5.398084598020214, −4.637139505641390, −4.049375134529482, −3.561529216578130, −2.822814028884509, −2.169009009815109, −1.772831373553119, −0.8092165996891768, 0, 0.8092165996891768, 1.772831373553119, 2.169009009815109, 2.822814028884509, 3.561529216578130, 4.049375134529482, 4.637139505641390, 5.398084598020214, 5.818713512225328, 6.135241241267493, 6.743863671908655, 7.610532397345585, 7.803459344228786, 8.423260656857132, 8.840453651728759, 9.416133815982729, 10.10739124478802, 10.38129071095207, 10.81138832099297, 11.29199528519012, 11.93718684392254, 12.40711149232765, 12.95508914753298, 13.48634255427193, 13.54810655100813

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