Properties

Label 2-236992-1.1-c1-0-39
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-s − 2·5-s + 7-s − 2·9-s − 13-s + 2·15-s + 8·17-s − 2·19-s − 21-s − 25-s + 5·27-s + 7·29-s − 7·31-s − 2·35-s − 2·37-s + 39-s − 11·41-s + 4·43-s + 4·45-s − 3·47-s + 49-s − 8·51-s + 6·53-s + 2·57-s + 4·59-s − 2·63-s + 2·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 0.377·7-s − 2/3·9-s − 0.277·13-s + 0.516·15-s + 1.94·17-s − 0.458·19-s − 0.218·21-s − 1/5·25-s + 0.962·27-s + 1.29·29-s − 1.25·31-s − 0.338·35-s − 0.328·37-s + 0.160·39-s − 1.71·41-s + 0.609·43-s + 0.596·45-s − 0.437·47-s + 1/7·49-s − 1.12·51-s + 0.824·53-s + 0.264·57-s + 0.520·59-s − 0.251·63-s + 0.248·65-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 \( 1 + T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
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

−12.95855262263903, −12.49654491914915, −12.01161465456721, −11.80998001308519, −11.49078877460552, −10.78175658032277, −10.44941578880924, −10.05593438076882, −9.421617594351384, −8.766599637508850, −8.412692069334811, −7.894116897362267, −7.592877524650943, −6.966433754560976, −6.490928423126186, −5.845485255306398, −5.394325637895372, −5.043819272616327, −4.461431714375956, −3.745834192336930, −3.412074260856361, −2.817362156597303, −2.086429892944347, −1.325730695423483, −0.6701208013831170, 0, 0.6701208013831170, 1.325730695423483, 2.086429892944347, 2.817362156597303, 3.412074260856361, 3.745834192336930, 4.461431714375956, 5.043819272616327, 5.394325637895372, 5.845485255306398, 6.490928423126186, 6.966433754560976, 7.592877524650943, 7.894116897362267, 8.412692069334811, 8.766599637508850, 9.421617594351384, 10.05593438076882, 10.44941578880924, 10.78175658032277, 11.49078877460552, 11.80998001308519, 12.01161465456721, 12.49654491914915, 12.95855262263903

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