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Properties

Label 2-259-259.181-c0-0-0
Degree $2$
Conductor $259$
Sign $-0.375 + 0.926i$
Analytic cond. $0.129257$
Root an. cond. $0.359524$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.266 − 1.50i)2-s + (−1.26 − 0.460i)4-s + (0.766 − 0.642i)7-s + (−0.266 + 0.460i)8-s + (−0.939 + 0.342i)9-s + (−0.766 + 1.32i)11-s + (−0.766 − 1.32i)14-s + (−0.407 − 0.342i)16-s + (0.266 + 1.50i)18-s + (1.79 + 1.50i)22-s + (0.939 + 1.62i)23-s + (0.173 − 0.984i)25-s + (−1.26 + 0.460i)28-s + (−0.173 + 0.300i)29-s + (−1.03 + 0.866i)32-s + ⋯
L(s)  = 1  + (0.266 − 1.50i)2-s + (−1.26 − 0.460i)4-s + (0.766 − 0.642i)7-s + (−0.266 + 0.460i)8-s + (−0.939 + 0.342i)9-s + (−0.766 + 1.32i)11-s + (−0.766 − 1.32i)14-s + (−0.407 − 0.342i)16-s + (0.266 + 1.50i)18-s + (1.79 + 1.50i)22-s + (0.939 + 1.62i)23-s + (0.173 − 0.984i)25-s + (−1.26 + 0.460i)28-s + (−0.173 + 0.300i)29-s + (−1.03 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 259 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.375 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.375 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(259\)    =    \(7 \cdot 37\)
Sign: $-0.375 + 0.926i$
Analytic conductor: \(0.129257\)
Root analytic conductor: \(0.359524\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{259} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 259,\ (\ :0),\ -0.375 + 0.926i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8169172844\)
\(L(\frac12)\) \(\approx\) \(0.8169172844\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.766 + 0.642i)T \)
37 \( 1 - T \)
good2 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
3 \( 1 + (0.939 - 0.342i)T^{2} \)
5 \( 1 + (-0.173 + 0.984i)T^{2} \)
11 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.766 - 0.642i)T^{2} \)
17 \( 1 + (-0.766 + 0.642i)T^{2} \)
19 \( 1 + (0.939 - 0.342i)T^{2} \)
23 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
41 \( 1 + (-0.766 - 0.642i)T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (-0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
83 \( 1 + (-0.766 + 0.642i)T^{2} \)
89 \( 1 + (-0.173 - 0.984i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−11.70658867770006844893663840227, −11.13138587149632602347344131505, −10.30982786934730222431763466544, −9.488354492087190788002444309843, −8.155630568260062339007399511115, −7.15494447985276402678409082636, −5.25150215346516643195060143445, −4.45438005960773995100990374074, −3.04231052666919472943502389079, −1.79620722045902103835867544304, 2.90648244531407647026125572368, 4.75444078955722594782466586454, 5.63446707641871008442846432135, 6.32369882085248349778769295928, 7.68274010415847466723106456206, 8.479994183587036104658490220724, 9.002440935745606874020215538542, 10.85015582352279559529404503396, 11.47985130075925360745009186685, 12.83064793335296590416286406617

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