Properties

Label 2-236992-1.1-c1-0-46
Degree $2$
Conductor $236992$
Sign $-1$
Analytic cond. $1892.39$
Root an. cond. $43.5016$
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  + 3-s − 2·5-s − 7-s − 2·9-s + 2·11-s + 13-s − 2·15-s + 6·19-s − 21-s − 25-s − 5·27-s − 29-s − 31-s + 2·33-s + 2·35-s − 6·37-s + 39-s + 3·41-s + 4·45-s + 3·47-s + 49-s + 6·53-s − 4·55-s + 6·57-s + 8·59-s − 10·61-s + 2·63-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 0.377·7-s − 2/3·9-s + 0.603·11-s + 0.277·13-s − 0.516·15-s + 1.37·19-s − 0.218·21-s − 1/5·25-s − 0.962·27-s − 0.185·29-s − 0.179·31-s + 0.348·33-s + 0.338·35-s − 0.986·37-s + 0.160·39-s + 0.468·41-s + 0.596·45-s + 0.437·47-s + 1/7·49-s + 0.824·53-s − 0.539·55-s + 0.794·57-s + 1.04·59-s − 1.28·61-s + 0.251·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(236992\)    =    \(2^{6} \cdot 7 \cdot 23^{2}\)
Sign: $-1$
Analytic conductor: \(1892.39\)
Root analytic conductor: \(43.5016\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{236992} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 236992,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
23 \( 1 \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−13.26143988813975, −12.59177298965122, −12.07574023669827, −11.73788403003714, −11.49275353707114, −10.81393464992706, −10.41887646951838, −9.728710198698621, −9.298252490906837, −8.916017917028031, −8.524960001226065, −7.828921475467155, −7.627786868544083, −7.107515638577163, −6.496632601458394, −5.982838381192041, −5.411426138338792, −4.984185717378922, −4.103336164164820, −3.777082775788056, −3.369002611284707, −2.839867963274763, −2.222722484593977, −1.453600443087633, −0.7435797800654889, 0, 0.7435797800654889, 1.453600443087633, 2.222722484593977, 2.839867963274763, 3.369002611284707, 3.777082775788056, 4.103336164164820, 4.984185717378922, 5.411426138338792, 5.982838381192041, 6.496632601458394, 7.107515638577163, 7.627786868544083, 7.828921475467155, 8.524960001226065, 8.916017917028031, 9.298252490906837, 9.728710198698621, 10.41887646951838, 10.81393464992706, 11.49275353707114, 11.73788403003714, 12.07574023669827, 12.59177298965122, 13.26143988813975

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