Properties

Label 2-236992-1.1-c1-0-49
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  − 5-s − 7-s − 3·9-s + 13-s − 2·17-s + 8·19-s − 4·25-s + 9·29-s + 4·31-s + 35-s + 6·37-s − 5·41-s + 8·43-s + 3·45-s + 49-s − 9·53-s − 5·61-s + 3·63-s − 65-s + 16·67-s − 12·71-s − 73-s + 4·79-s + 9·81-s + 12·83-s + 2·85-s + 9·89-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 9-s + 0.277·13-s − 0.485·17-s + 1.83·19-s − 4/5·25-s + 1.67·29-s + 0.718·31-s + 0.169·35-s + 0.986·37-s − 0.780·41-s + 1.21·43-s + 0.447·45-s + 1/7·49-s − 1.23·53-s − 0.640·61-s + 0.377·63-s − 0.124·65-s + 1.95·67-s − 1.42·71-s − 0.117·73-s + 0.450·79-s + 81-s + 1.31·83-s + 0.216·85-s + 0.953·89-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 \( 1 + p T^{2} \)
5 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−13.25962294371044, −12.55986423540200, −12.07284533618275, −11.75590854667065, −11.41409279495128, −10.86864027026778, −10.39900897036371, −9.782578897212401, −9.418373491613078, −8.963956358951285, −8.376303316214133, −7.919291592922813, −7.649042886873763, −6.914608278453018, −6.425161107681398, −5.984898261309234, −5.529892978097495, −4.786038667360808, −4.564642270241395, −3.542187371023166, −3.468219889074402, −2.666207724778865, −2.371059850626623, −1.278152193217244, −0.7849369451980740, 0, 0.7849369451980740, 1.278152193217244, 2.371059850626623, 2.666207724778865, 3.468219889074402, 3.542187371023166, 4.564642270241395, 4.786038667360808, 5.529892978097495, 5.984898261309234, 6.425161107681398, 6.914608278453018, 7.649042886873763, 7.919291592922813, 8.376303316214133, 8.963956358951285, 9.418373491613078, 9.782578897212401, 10.39900897036371, 10.86864027026778, 11.41409279495128, 11.75590854667065, 12.07284533618275, 12.55986423540200, 13.25962294371044

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