Properties

Label 2-236992-1.1-c1-0-42
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  − 2·3-s − 2·5-s − 7-s + 9-s + 2·11-s + 4·13-s + 4·15-s + 6·17-s + 2·21-s − 25-s + 4·27-s + 2·29-s − 4·31-s − 4·33-s + 2·35-s − 8·39-s + 6·41-s − 6·43-s − 2·45-s + 49-s − 12·51-s − 12·53-s − 4·55-s − 10·59-s + 2·61-s − 63-s − 8·65-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s + 1.10·13-s + 1.03·15-s + 1.45·17-s + 0.436·21-s − 1/5·25-s + 0.769·27-s + 0.371·29-s − 0.718·31-s − 0.696·33-s + 0.338·35-s − 1.28·39-s + 0.937·41-s − 0.914·43-s − 0.298·45-s + 1/7·49-s − 1.68·51-s − 1.64·53-s − 0.539·55-s − 1.30·59-s + 0.256·61-s − 0.125·63-s − 0.992·65-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 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−12.91514929648448, −12.51830898871050, −12.10528947107054, −11.74686897665321, −11.35503318424093, −10.89790754852444, −10.54842825497156, −9.975336337214937, −9.414092098853447, −8.980260795127164, −8.365133541977800, −7.842738374361592, −7.553840182418126, −6.813046092111488, −6.338129081431323, −6.055139626484727, −5.504300154044657, −4.998437472739846, −4.392353098083263, −3.853238850212310, −3.344139739187297, −3.017075507387276, −1.897641630118187, −1.230040224048262, −0.7093854012051795, 0, 0.7093854012051795, 1.230040224048262, 1.897641630118187, 3.017075507387276, 3.344139739187297, 3.853238850212310, 4.392353098083263, 4.998437472739846, 5.504300154044657, 6.055139626484727, 6.338129081431323, 6.813046092111488, 7.553840182418126, 7.842738374361592, 8.365133541977800, 8.980260795127164, 9.414092098853447, 9.975336337214937, 10.54842825497156, 10.89790754852444, 11.35503318424093, 11.74686897665321, 12.10528947107054, 12.51830898871050, 12.91514929648448

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