Properties

Label 2-236992-1.1-c1-0-29
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 + 4·5-s − 7-s + 9-s + 4·11-s + 2·13-s − 8·15-s + 4·17-s + 2·21-s + 11·25-s + 4·27-s − 2·29-s − 6·31-s − 8·33-s − 4·35-s + 2·37-s − 4·39-s − 6·41-s − 4·43-s + 4·45-s − 10·47-s + 49-s − 8·51-s + 6·53-s + 16·55-s + 2·59-s + 8·61-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.78·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 2.06·15-s + 0.970·17-s + 0.436·21-s + 11/5·25-s + 0.769·27-s − 0.371·29-s − 1.07·31-s − 1.39·33-s − 0.676·35-s + 0.328·37-s − 0.640·39-s − 0.937·41-s − 0.609·43-s + 0.596·45-s − 1.45·47-s + 1/7·49-s − 1.12·51-s + 0.824·53-s + 2.15·55-s + 0.260·59-s + 1.02·61-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

\(L(1)\) \(\approx\) \(3.022114105\)
\(L(\frac12)\) \(\approx\) \(3.022114105\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−13.04259831892187, −12.40381963154546, −11.94318538009811, −11.48327465731709, −11.13204996811398, −10.45570101787282, −10.13697527760935, −9.788295495090892, −9.264738579505073, −8.793822006283349, −8.445048631512720, −7.475917976878262, −6.999119240361587, −6.434979166036902, −6.176862549658708, −5.831455946857182, −5.273079122772882, −4.982633745352145, −4.226701020096595, −3.391872321517632, −3.190806926314502, −2.136610034893200, −1.731791774263759, −1.136819866075821, −0.5596947645003818, 0.5596947645003818, 1.136819866075821, 1.731791774263759, 2.136610034893200, 3.190806926314502, 3.391872321517632, 4.226701020096595, 4.982633745352145, 5.273079122772882, 5.831455946857182, 6.176862549658708, 6.434979166036902, 6.999119240361587, 7.475917976878262, 8.445048631512720, 8.793822006283349, 9.264738579505073, 9.788295495090892, 10.13697527760935, 10.45570101787282, 11.13204996811398, 11.48327465731709, 11.94318538009811, 12.40381963154546, 13.04259831892187

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