Properties

Label 2-260-260.199-c0-0-0
Degree $2$
Conductor $260$
Sign $0.859 - 0.511i$
Analytic cond. $0.129756$
Root an. cond. $0.360217$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999·8-s + (0.5 + 0.866i)9-s + (0.499 + 0.866i)10-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)16-s + (−1.5 + 0.866i)17-s − 0.999·18-s − 0.999·20-s + (−0.499 − 0.866i)25-s + (0.499 + 0.866i)26-s + (−0.5 + 0.866i)29-s + (−0.499 − 0.866i)32-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999·8-s + (0.5 + 0.866i)9-s + (0.499 + 0.866i)10-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)16-s + (−1.5 + 0.866i)17-s − 0.999·18-s − 0.999·20-s + (−0.499 − 0.866i)25-s + (0.499 + 0.866i)26-s + (−0.5 + 0.866i)29-s + (−0.499 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(0.129756\)
Root analytic conductor: \(0.360217\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :0),\ 0.859 - 0.511i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6224374348\)
\(L(\frac12)\) \(\approx\) \(0.6224374348\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.73iT - T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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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

−12.70843432330213825977163120907, −10.92964267101808204889946191360, −10.32861644157476669695911839420, −9.153176252344279748857334196517, −8.473277589334956967593091506391, −7.52719041942000516695662327169, −6.27672462645822086311697588438, −5.31678348172953065925492238190, −4.33426405051252463644674277634, −1.75313437929249790249907423913, 1.96535227925586864006146446441, 3.33248629198086177599125501314, 4.53003450634788901396152684989, 6.42410185191117863987556631355, 7.14700182485008112197213593774, 8.606974884052520724801682866152, 9.506389038712524404579253548434, 10.13388902225517093990909170710, 11.36956363915845739728229405761, 11.66955307630825582851493408218

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