Properties

Label 2-116242-1.1-c1-0-24
Degree $2$
Conductor $116242$
Sign $-1$
Analytic cond. $928.197$
Root an. cond. $30.4663$
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 + 3-s + 4-s + 3·5-s + 6-s − 7-s + 8-s − 2·9-s + 3·10-s − 11-s + 12-s − 2·13-s − 14-s + 3·15-s + 16-s − 2·17-s − 2·18-s + 3·20-s − 21-s − 22-s + 23-s + 24-s + 4·25-s − 2·26-s − 5·27-s − 28-s + 3·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.948·10-s − 0.301·11-s + 0.288·12-s − 0.554·13-s − 0.267·14-s + 0.774·15-s + 1/4·16-s − 0.485·17-s − 0.471·18-s + 0.670·20-s − 0.218·21-s − 0.213·22-s + 0.208·23-s + 0.204·24-s + 4/5·25-s − 0.392·26-s − 0.962·27-s − 0.188·28-s + 0.557·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(116242\)    =    \(2 \cdot 7 \cdot 19^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(928.197\)
Root analytic conductor: \(30.4663\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{116242} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 116242,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 + T \)
19 \( 1 \)
23 \( 1 - T \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 - 5 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 - 13 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 15 T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 + 13 T + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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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

−13.77132073706026, −13.30743054991375, −13.22762324996899, −12.50304414792681, −12.03705606116873, −11.51184386596189, −10.78226974353819, −10.56018443888712, −9.830324715838805, −9.489800226790010, −9.017064221731680, −8.475853961670974, −7.916647896844029, −7.214476522571863, −6.810926081427830, −6.174999830051988, −5.736865007877581, −5.306398496235606, −4.810558959375549, −4.004767909066793, −3.469863260567580, −2.744689779959434, −2.393968976212316, −1.984707384195488, −1.073114855899593, 0, 1.073114855899593, 1.984707384195488, 2.393968976212316, 2.744689779959434, 3.469863260567580, 4.004767909066793, 4.810558959375549, 5.306398496235606, 5.736865007877581, 6.174999830051988, 6.810926081427830, 7.214476522571863, 7.916647896844029, 8.475853961670974, 9.017064221731680, 9.489800226790010, 9.830324715838805, 10.56018443888712, 10.78226974353819, 11.51184386596189, 12.03705606116873, 12.50304414792681, 13.22762324996899, 13.30743054991375, 13.77132073706026

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