Properties

Label 2-236992-1.1-c1-0-50
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  + 7-s − 3·9-s + 4·11-s + 2·13-s + 4·17-s − 4·19-s − 5·25-s − 2·29-s + 4·31-s − 4·37-s − 6·41-s + 4·43-s + 4·47-s + 49-s − 12·53-s − 8·59-s − 3·63-s − 4·67-s + 8·71-s − 10·73-s + 4·77-s − 8·79-s + 9·81-s + 12·83-s − 4·89-s + 2·91-s + 12·97-s + ⋯
L(s)  = 1  + 0.377·7-s − 9-s + 1.20·11-s + 0.554·13-s + 0.970·17-s − 0.917·19-s − 25-s − 0.371·29-s + 0.718·31-s − 0.657·37-s − 0.937·41-s + 0.609·43-s + 0.583·47-s + 1/7·49-s − 1.64·53-s − 1.04·59-s − 0.377·63-s − 0.488·67-s + 0.949·71-s − 1.17·73-s + 0.455·77-s − 0.900·79-s + 81-s + 1.31·83-s − 0.423·89-s + 0.209·91-s + 1.21·97-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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−13.19720287603442, −12.49045043701971, −12.19272920578576, −11.65352661065101, −11.44050352966906, −10.82517352814053, −10.45058198706354, −9.859426581431601, −9.250759382192965, −9.010967793525084, −8.333999353845320, −8.135667091263667, −7.524899929093115, −6.942315018217133, −6.278840310121873, −6.050283007716587, −5.570253790903216, −4.895235492538888, −4.368679337960943, −3.787507852992956, −3.349385512049100, −2.799514788502456, −1.937991269931837, −1.578118990442126, −0.8127673865331493, 0, 0.8127673865331493, 1.578118990442126, 1.937991269931837, 2.799514788502456, 3.349385512049100, 3.787507852992956, 4.368679337960943, 4.895235492538888, 5.570253790903216, 6.050283007716587, 6.278840310121873, 6.942315018217133, 7.524899929093115, 8.135667091263667, 8.333999353845320, 9.010967793525084, 9.250759382192965, 9.859426581431601, 10.45058198706354, 10.82517352814053, 11.44050352966906, 11.65352661065101, 12.19272920578576, 12.49045043701971, 13.19720287603442

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