Properties

Label 2-259182-1.1-c1-0-111
Degree $2$
Conductor $259182$
Sign $-1$
Analytic cond. $2069.57$
Root an. cond. $45.4926$
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  + 2-s + 4-s − 3·5-s + 7-s + 8-s − 3·10-s + 2·13-s + 14-s + 16-s + 17-s − 5·19-s − 3·20-s − 3·23-s + 4·25-s + 2·26-s + 28-s + 4·29-s + 31-s + 32-s + 34-s − 3·35-s − 37-s − 5·38-s − 3·40-s + 2·41-s + 4·43-s − 3·46-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.377·7-s + 0.353·8-s − 0.948·10-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 1.14·19-s − 0.670·20-s − 0.625·23-s + 4/5·25-s + 0.392·26-s + 0.188·28-s + 0.742·29-s + 0.179·31-s + 0.176·32-s + 0.171·34-s − 0.507·35-s − 0.164·37-s − 0.811·38-s − 0.474·40-s + 0.312·41-s + 0.609·43-s − 0.442·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(259182\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(2069.57\)
Root analytic conductor: \(45.4926\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{259182} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 259182,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
3 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
17 \( 1 - T \)
good5 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−12.89167760539821, −12.57982131780310, −12.10668154763274, −11.73596973160255, −11.31871144028374, −10.92176302245441, −10.37025736271758, −10.09046759002798, −9.215252639222435, −8.656521903635751, −8.325631838248636, −7.890326342343377, −7.397268041495766, −6.950950690701182, −6.345455130382124, −5.951618161525606, −5.317495749095145, −4.748457106694981, −4.242098828854033, −3.899660292002681, −3.521219249152777, −2.706316827204557, −2.301144210695468, −1.464299282371576, −0.8059212259681444, 0, 0.8059212259681444, 1.464299282371576, 2.301144210695468, 2.706316827204557, 3.521219249152777, 3.899660292002681, 4.242098828854033, 4.748457106694981, 5.317495749095145, 5.951618161525606, 6.345455130382124, 6.950950690701182, 7.397268041495766, 7.890326342343377, 8.325631838248636, 8.656521903635751, 9.215252639222435, 10.09046759002798, 10.37025736271758, 10.92176302245441, 11.31871144028374, 11.73596973160255, 12.10668154763274, 12.57982131780310, 12.89167760539821

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