Properties

Label 2-236992-1.1-c1-0-66
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 7-s + 9-s + 4·11-s − 6·17-s − 6·19-s + 2·21-s − 5·25-s + 4·27-s − 10·29-s + 4·31-s − 8·33-s − 2·37-s − 10·41-s − 4·43-s + 12·47-s + 49-s + 12·51-s − 6·53-s + 12·57-s + 2·59-s − 63-s − 8·71-s − 6·73-s + 10·75-s − 4·77-s + 8·79-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 1.45·17-s − 1.37·19-s + 0.436·21-s − 25-s + 0.769·27-s − 1.85·29-s + 0.718·31-s − 1.39·33-s − 0.328·37-s − 1.56·41-s − 0.609·43-s + 1.75·47-s + 1/7·49-s + 1.68·51-s − 0.824·53-s + 1.58·57-s + 0.260·59-s − 0.125·63-s − 0.949·71-s − 0.702·73-s + 1.15·75-s − 0.455·77-s + 0.900·79-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: \(2\)
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 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.35776376807183, −12.82452965494936, −12.36456500142421, −11.82356565566345, −11.54336261783736, −11.19288594863172, −10.59185998470640, −10.31211753461185, −9.678602475454280, −9.101459377068378, −8.787600269468704, −8.349761997738398, −7.549364447662209, −6.991094123789421, −6.621839589691697, −6.181469975690306, −5.880887334287476, −5.261769380491527, −4.594455973691868, −4.229801905394361, −3.732446867168440, −3.099633773966796, −2.125244682204795, −1.889944151641182, −1.013656661566588, 0, 0, 1.013656661566588, 1.889944151641182, 2.125244682204795, 3.099633773966796, 3.732446867168440, 4.229801905394361, 4.594455973691868, 5.261769380491527, 5.880887334287476, 6.181469975690306, 6.621839589691697, 6.991094123789421, 7.549364447662209, 8.349761997738398, 8.787600269468704, 9.101459377068378, 9.678602475454280, 10.31211753461185, 10.59185998470640, 11.19288594863172, 11.54336261783736, 11.82356565566345, 12.36456500142421, 12.82452965494936, 13.35776376807183

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