Properties

Label 2-236992-1.1-c1-0-31
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·5-s + 7-s + 9-s + 2·13-s + 4·15-s + 2·17-s + 4·19-s − 2·21-s − 25-s + 4·27-s + 2·29-s + 10·31-s − 2·35-s + 8·37-s − 4·39-s + 2·41-s + 4·43-s − 2·45-s + 6·47-s + 49-s − 4·51-s − 12·53-s − 8·57-s + 10·59-s − 6·61-s + 63-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.554·13-s + 1.03·15-s + 0.485·17-s + 0.917·19-s − 0.436·21-s − 1/5·25-s + 0.769·27-s + 0.371·29-s + 1.79·31-s − 0.338·35-s + 1.31·37-s − 0.640·39-s + 0.312·41-s + 0.609·43-s − 0.298·45-s + 0.875·47-s + 1/7·49-s − 0.560·51-s − 1.64·53-s − 1.05·57-s + 1.30·59-s − 0.768·61-s + 0.125·63-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: \(0\)
Selberg data: \((2,\ 236992,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.907104590\)
\(L(\frac12)\) \(\approx\) \(1.907104590\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−12.81653971037072, −12.14984472436954, −11.93816887922578, −11.51392972097365, −11.19728391942426, −10.79442726967678, −10.12025299140415, −9.871731490695512, −9.154722527971073, −8.551911606608587, −8.181514957315934, −7.540809722153790, −7.421787956801257, −6.556226408482391, −6.101571030574674, −5.852818588892983, −5.107734623662331, −4.728640746625910, −4.250152229185730, −3.677202594608136, −3.025574702187503, −2.529841341805109, −1.538384446092606, −0.8263635785516721, −0.5925402483997291, 0.5925402483997291, 0.8263635785516721, 1.538384446092606, 2.529841341805109, 3.025574702187503, 3.677202594608136, 4.250152229185730, 4.728640746625910, 5.107734623662331, 5.852818588892983, 6.101571030574674, 6.556226408482391, 7.421787956801257, 7.540809722153790, 8.181514957315934, 8.551911606608587, 9.154722527971073, 9.871731490695512, 10.12025299140415, 10.79442726967678, 11.19728391942426, 11.51392972097365, 11.93816887922578, 12.14984472436954, 12.81653971037072

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