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Properties

Label 2-260-260.199-c0-0-1
Degree $2$
Conductor $260$
Sign $0.0128 + 0.999i$
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 − 0.999·10-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s + (1.5 − 0.866i)17-s + 0.999·18-s + (−0.499 + 0.866i)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 − 0.999·10-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s + (1.5 − 0.866i)17-s + 0.999·18-s + (−0.499 + 0.866i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.0128 + 0.999i$
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: \((2,\ 260,\ (\ :0),\ 0.0128 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8456998000\)
\(L(\frac12)\) \(\approx\) \(0.8456998000\)
\(L(1)\) 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

−12.04934332150364539544400017173, −11.28573325298416855138741011212, −10.12057210152045458047987899766, −9.376869060991366195821513888207, −8.240700756301197017756863509746, −7.06573743090012430937731930116, −5.31247066409717182369300541030, −4.73268209213614871980307210035, −3.44221126870526593017649212231, −1.68812179178659041672434700093, 3.14764854362747282878806641578, 4.00718567353546154410126621653, 5.52739307835739150254260746983, 6.50574104872451082874333522878, 7.48120611764364458800286102176, 8.164153071975824916867905921930, 9.580964214640746294850821197644, 10.47493586434217829683204173858, 11.89912388242201907344183249686, 12.41378958102965389400848747775

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