Properties

Label 2-236992-1.1-c1-0-33
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  − 3·3-s − 4·5-s − 7-s + 6·9-s − 2·11-s − 5·13-s + 12·15-s + 4·19-s + 3·21-s + 11·25-s − 9·27-s + 3·29-s − 5·31-s + 6·33-s + 4·35-s + 4·37-s + 15·39-s + 5·41-s + 4·43-s − 24·45-s + 11·47-s + 49-s + 8·55-s − 12·57-s − 12·59-s − 6·61-s − 6·63-s + ⋯
L(s)  = 1  − 1.73·3-s − 1.78·5-s − 0.377·7-s + 2·9-s − 0.603·11-s − 1.38·13-s + 3.09·15-s + 0.917·19-s + 0.654·21-s + 11/5·25-s − 1.73·27-s + 0.557·29-s − 0.898·31-s + 1.04·33-s + 0.676·35-s + 0.657·37-s + 2.40·39-s + 0.780·41-s + 0.609·43-s − 3.57·45-s + 1.60·47-s + 1/7·49-s + 1.07·55-s − 1.58·57-s − 1.56·59-s − 0.768·61-s − 0.755·63-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 \( 1 + p T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−12.66846099849748, −12.43947960235583, −12.25201254834021, −11.82222161804091, −11.24358431431917, −10.95035758410219, −10.61977393957603, −10.01902700993255, −9.489067330088911, −9.077774562259118, −8.195771314454801, −7.714587431254635, −7.395589337289032, −7.078581314765821, −6.518522264976075, −5.862683992681113, −5.399881215929885, −4.917205266785068, −4.470912513039820, −4.097327636967153, −3.375257146233547, −2.837904125040052, −2.095035792515886, −0.9491904886868224, −0.6037938407871409, 0, 0.6037938407871409, 0.9491904886868224, 2.095035792515886, 2.837904125040052, 3.375257146233547, 4.097327636967153, 4.470912513039820, 4.917205266785068, 5.399881215929885, 5.862683992681113, 6.518522264976075, 7.078581314765821, 7.395589337289032, 7.714587431254635, 8.195771314454801, 9.077774562259118, 9.489067330088911, 10.01902700993255, 10.61977393957603, 10.95035758410219, 11.24358431431917, 11.82222161804091, 12.25201254834021, 12.43947960235583, 12.66846099849748

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