Properties

Label 2-259182-1.1-c1-0-115
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 − 4·5-s + 7-s − 8-s + 4·10-s + 6·13-s − 14-s + 16-s + 17-s + 6·19-s − 4·20-s − 4·23-s + 11·25-s − 6·26-s + 28-s + 6·29-s − 8·31-s − 32-s − 34-s − 4·35-s − 2·37-s − 6·38-s + 4·40-s + 6·41-s + 6·43-s + 4·46-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.78·5-s + 0.377·7-s − 0.353·8-s + 1.26·10-s + 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s + 1.37·19-s − 0.894·20-s − 0.834·23-s + 11/5·25-s − 1.17·26-s + 0.188·28-s + 1.11·29-s − 1.43·31-s − 0.176·32-s − 0.171·34-s − 0.676·35-s − 0.328·37-s − 0.973·38-s + 0.632·40-s + 0.937·41-s + 0.914·43-s + 0.589·46-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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−12.85315327267324, −12.24009977957019, −12.16537095405986, −11.55924557305961, −11.18755569628835, −10.76900523566471, −10.54289443720772, −9.760780010682023, −9.092397097714866, −8.884876188781507, −8.311505664382920, −7.852618632806142, −7.620332079910548, −7.162138902477525, −6.585979286429652, −5.876831818641115, −5.598934030902745, −4.722102028931709, −4.254601453307947, −3.748584093138221, −3.291812684079863, −2.858916110310088, −1.904138039365034, −1.169861202220025, −0.8200919312924066, 0, 0.8200919312924066, 1.169861202220025, 1.904138039365034, 2.858916110310088, 3.291812684079863, 3.748584093138221, 4.254601453307947, 4.722102028931709, 5.598934030902745, 5.876831818641115, 6.585979286429652, 7.162138902477525, 7.620332079910548, 7.852618632806142, 8.311505664382920, 8.884876188781507, 9.092397097714866, 9.760780010682023, 10.54289443720772, 10.76900523566471, 11.18755569628835, 11.55924557305961, 12.16537095405986, 12.24009977957019, 12.85315327267324

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