Properties

Label 2-236992-1.1-c1-0-25
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  − 3-s − 4·5-s − 7-s − 2·9-s − 2·13-s + 4·15-s − 5·17-s + 7·19-s + 21-s + 11·25-s + 5·27-s − 8·29-s + 2·31-s + 4·35-s + 8·37-s + 2·39-s − 2·41-s + 3·43-s + 8·45-s + 49-s + 5·51-s − 6·53-s − 7·57-s − 9·59-s + 6·61-s + 2·63-s + 8·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.78·5-s − 0.377·7-s − 2/3·9-s − 0.554·13-s + 1.03·15-s − 1.21·17-s + 1.60·19-s + 0.218·21-s + 11/5·25-s + 0.962·27-s − 1.48·29-s + 0.359·31-s + 0.676·35-s + 1.31·37-s + 0.320·39-s − 0.312·41-s + 0.457·43-s + 1.19·45-s + 1/7·49-s + 0.700·51-s − 0.824·53-s − 0.927·57-s − 1.17·59-s + 0.768·61-s + 0.251·63-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: Trivial
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 \( 1 + T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 17 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 13 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−12.98170281297260, −12.53670892082551, −12.03671226654489, −11.75524390154001, −11.34166520736149, −10.91191692155566, −10.71831471823167, −9.773964108527602, −9.328284778598541, −9.040186617435747, −8.231289899996298, −7.949869688623839, −7.502393984615279, −6.998353363237901, −6.560044154091428, −5.976492161721512, −5.347852107301287, −4.891006491431573, −4.434005620707213, −3.813148742330429, −3.379601606708579, −2.825756389712174, −2.276049985126957, −1.190053981617121, −0.5187472618384308, 0, 0.5187472618384308, 1.190053981617121, 2.276049985126957, 2.825756389712174, 3.379601606708579, 3.813148742330429, 4.434005620707213, 4.891006491431573, 5.347852107301287, 5.976492161721512, 6.560044154091428, 6.998353363237901, 7.502393984615279, 7.949869688623839, 8.231289899996298, 9.040186617435747, 9.328284778598541, 9.773964108527602, 10.71831471823167, 10.91191692155566, 11.34166520736149, 11.75524390154001, 12.03671226654489, 12.53670892082551, 12.98170281297260

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