Properties

Label 2-236992-1.1-c1-0-32
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

−13.11603905502589, −12.69608607368191, −12.09448026628069, −11.69497506003868, −11.33995971244099, −10.86887901948019, −10.42362453406268, −10.02565710496589, −9.190534037434870, −8.891994360272644, −8.386449911204058, −8.036956964250452, −7.681445761999329, −6.786569965713322, −6.492918690931907, −6.035702617345927, −5.480065073816194, −4.781674996877597, −4.432880135385412, −3.935952004944612, −3.188308423165901, −2.814971829516558, −2.019778964959231, −1.649714884664109, −0.5932125666133799, 0, 0.5932125666133799, 1.649714884664109, 2.019778964959231, 2.814971829516558, 3.188308423165901, 3.935952004944612, 4.432880135385412, 4.781674996877597, 5.480065073816194, 6.035702617345927, 6.492918690931907, 6.786569965713322, 7.681445761999329, 8.036956964250452, 8.386449911204058, 8.891994360272644, 9.190534037434870, 10.02565710496589, 10.42362453406268, 10.86887901948019, 11.33995971244099, 11.69497506003868, 12.09448026628069, 12.69608607368191, 13.11603905502589

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