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Properties

Label 2-259-259.258-c0-0-0
Degree $2$
Conductor $259$
Sign $1$
Analytic cond. $0.129257$
Root an. cond. $0.359524$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 7-s + 9-s − 2·11-s + 16-s − 25-s − 28-s + 36-s − 37-s − 2·44-s + 49-s − 2·53-s − 63-s + 64-s + 2·67-s + 2·71-s + 2·77-s + 81-s − 2·99-s − 100-s + 2·107-s − 112-s + ⋯
L(s)  = 1  + 4-s − 7-s + 9-s − 2·11-s + 16-s − 25-s − 28-s + 36-s − 37-s − 2·44-s + 49-s − 2·53-s − 63-s + 64-s + 2·67-s + 2·71-s + 2·77-s + 81-s − 2·99-s − 100-s + 2·107-s − 112-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 259 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(259\)    =    \(7 \cdot 37\)
Sign: $1$
Analytic conductor: \(0.129257\)
Root analytic conductor: \(0.359524\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{259} (258, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 259,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8177623440\)
\(L(\frac12)\) \(\approx\) \(0.8177623440\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + T \)
37 \( 1 + T \)
good2 \( ( 1 - T )( 1 + T ) \)
3 \( ( 1 - T )( 1 + T ) \)
5 \( 1 + T^{2} \)
11 \( ( 1 + T )^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 + T )^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( ( 1 - T )^{2} \)
71 \( ( 1 - T )^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( 1 + T^{2} \)
97 \( 1 + T^{2} \)
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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

−12.46724060907624933811048361629, −11.13827325994637947077672697939, −10.30045467050284559881481888981, −9.720405567998881370383886140334, −8.058411393480327882020630438450, −7.29206301124526335201996856236, −6.32869690945378355499677776088, −5.18797716022843370477460667730, −3.47726054522894575174849889748, −2.25422264825589549742876672506, 2.25422264825589549742876672506, 3.47726054522894575174849889748, 5.18797716022843370477460667730, 6.32869690945378355499677776088, 7.29206301124526335201996856236, 8.058411393480327882020630438450, 9.720405567998881370383886140334, 10.30045467050284559881481888981, 11.13827325994637947077672697939, 12.46724060907624933811048361629

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