Properties

Label 2-236992-1.1-c1-0-22
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 3·5-s + 7-s + 9-s + 2·11-s + 3·13-s − 6·15-s − 6·17-s + 2·21-s + 4·25-s − 4·27-s − 29-s + 2·31-s + 4·33-s − 3·35-s + 10·37-s + 6·39-s − 41-s − 2·43-s − 3·45-s + 6·47-s + 49-s − 12·51-s + 9·53-s − 6·55-s + 4·59-s − 15·61-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.34·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.832·13-s − 1.54·15-s − 1.45·17-s + 0.436·21-s + 4/5·25-s − 0.769·27-s − 0.185·29-s + 0.359·31-s + 0.696·33-s − 0.507·35-s + 1.64·37-s + 0.960·39-s − 0.156·41-s − 0.304·43-s − 0.447·45-s + 0.875·47-s + 1/7·49-s − 1.68·51-s + 1.23·53-s − 0.809·55-s + 0.520·59-s − 1.92·61-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: \(0\)
Selberg data: \((2,\ 236992,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.004970636\)
\(L(\frac12)\) \(\approx\) \(3.004970636\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 15 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 17 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−13.16273104746629, −12.38605652387790, −11.83158057553060, −11.59622625205634, −11.06998690626395, −10.76388310842553, −10.10317140239198, −9.354126601194907, −8.970001315877417, −8.754561763434097, −8.170825964379387, −7.825501391115974, −7.442121626572887, −6.841339920460746, −6.285284196222209, −5.839259998987549, −4.920703788723838, −4.436263371915949, −4.012442491769120, −3.627482544494926, −3.117100150740619, −2.398194373826455, −2.004385273939844, −1.121328087768335, −0.4761950570377571, 0.4761950570377571, 1.121328087768335, 2.004385273939844, 2.398194373826455, 3.117100150740619, 3.627482544494926, 4.012442491769120, 4.436263371915949, 4.920703788723838, 5.839259998987549, 6.285284196222209, 6.841339920460746, 7.442121626572887, 7.825501391115974, 8.170825964379387, 8.754561763434097, 8.970001315877417, 9.354126601194907, 10.10317140239198, 10.76388310842553, 11.06998690626395, 11.59622625205634, 11.83158057553060, 12.38605652387790, 13.16273104746629

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