Properties

Label 2-236992-1.1-c1-0-45
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 + 2·13-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: \(1\)
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 \( 1 + p T^{2} \)
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

−13.31671381820014, −12.54058150300708, −12.19708236577316, −11.67109800404728, −11.42315671855939, −10.61394452333568, −10.42494952135043, −10.13316159986352, −9.492840777488622, −8.895981795335713, −8.513117550803300, −8.113491791429640, −7.169267902416785, −6.724862921824559, −6.472566647890210, −5.904330987482616, −5.560630397657538, −5.140901509760719, −4.342212247920921, −4.121262407489313, −3.232061757508022, −2.594051986731048, −2.121428956821000, −1.343305309803312, −0.7373708693721311, 0, 0.7373708693721311, 1.343305309803312, 2.121428956821000, 2.594051986731048, 3.232061757508022, 4.121262407489313, 4.342212247920921, 5.140901509760719, 5.560630397657538, 5.904330987482616, 6.472566647890210, 6.724862921824559, 7.169267902416785, 8.113491791429640, 8.513117550803300, 8.895981795335713, 9.492840777488622, 10.13316159986352, 10.42494952135043, 10.61394452333568, 11.42315671855939, 11.67109800404728, 12.19708236577316, 12.54058150300708, 13.31671381820014

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