Properties

Label 2-236992-1.1-c1-0-34
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  + 3-s + 7-s − 2·9-s + 6·11-s + 3·13-s + 21-s − 5·25-s − 5·27-s + 3·29-s + 7·31-s + 6·33-s + 8·37-s + 3·39-s − 11·41-s − 4·43-s − 47-s + 49-s + 4·53-s + 12·59-s − 6·61-s − 2·63-s − 12·67-s + 5·71-s + 15·73-s − 5·75-s + 6·77-s − 4·79-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s − 2/3·9-s + 1.80·11-s + 0.832·13-s + 0.218·21-s − 25-s − 0.962·27-s + 0.557·29-s + 1.25·31-s + 1.04·33-s + 1.31·37-s + 0.480·39-s − 1.71·41-s − 0.609·43-s − 0.145·47-s + 1/7·49-s + 0.549·53-s + 1.56·59-s − 0.768·61-s − 0.251·63-s − 1.46·67-s + 0.593·71-s + 1.75·73-s − 0.577·75-s + 0.683·77-s − 0.450·79-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

\(L(1)\) \(\approx\) \(4.601823935\)
\(L(\frac12)\) \(\approx\) \(4.601823935\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 - 15 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−13.16085345938969, −12.16731164338023, −11.97541340882803, −11.47057719191428, −11.31910208140253, −10.57487364603463, −9.957803616471321, −9.647615527193484, −9.048507429783530, −8.586016718588237, −8.405346481410515, −7.838326391876798, −7.265468121065500, −6.593021847338203, −6.224648078238417, −5.914939495055068, −5.118431055155884, −4.561463735762669, −4.033413580080068, −3.488245496080828, −3.201382518712128, −2.300551909717164, −1.874907429103354, −1.166727420307284, −0.6121720088005784, 0.6121720088005784, 1.166727420307284, 1.874907429103354, 2.300551909717164, 3.201382518712128, 3.488245496080828, 4.033413580080068, 4.561463735762669, 5.118431055155884, 5.914939495055068, 6.224648078238417, 6.593021847338203, 7.265468121065500, 7.838326391876798, 8.405346481410515, 8.586016718588237, 9.048507429783530, 9.647615527193484, 9.957803616471321, 10.57487364603463, 11.31910208140253, 11.47057719191428, 11.97541340882803, 12.16731164338023, 13.16085345938969

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