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Properties

Label 2-260-260.63-c0-0-0
Degree $2$
Conductor $260$
Sign $0.492 - 0.870i$
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 + (0.866 − 0.5i)5-s − 0.999·8-s + (−0.866 − 0.5i)9-s + (0.866 + 0.499i)10-s + (−0.866 + 0.5i)13-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.133i)17-s − 0.999i·18-s + 0.999i·20-s + (0.499 − 0.866i)25-s + (−0.866 − 0.499i)26-s + (0.866 − 0.5i)29-s + (0.499 − 0.866i)32-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.866 − 0.5i)5-s − 0.999·8-s + (−0.866 − 0.5i)9-s + (0.866 + 0.499i)10-s + (−0.866 + 0.5i)13-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.133i)17-s − 0.999i·18-s + 0.999i·20-s + (0.499 − 0.866i)25-s + (−0.866 − 0.499i)26-s + (0.866 − 0.5i)29-s + (0.499 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.492 - 0.870i$
Analytic conductor: \(0.129756\)
Root analytic conductor: \(0.360217\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :0),\ 0.492 - 0.870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9234907663\)
\(L(\frac12)\) \(\approx\) \(0.9234907663\)
\(L(1)\) 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 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (0.866 - 0.5i)T \)
good3 \( 1 + (0.866 + 0.5i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
59 \( 1 + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.866 + 0.5i)T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
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

−12.38812295064890636517140429762, −11.89983803995641904711793188630, −10.32766289793960705831716044542, −9.109967366146432432919446554193, −8.663720461668544916473436709978, −7.26095671173028544333470248470, −6.23776569963208445536544672013, −5.39839195160831389465764793075, −4.30411632531270821362541913488, −2.64659054785408095935918971675, 2.17922601816848025293999803865, 3.14749377840576175714827016313, 4.88848868366358079334351403999, 5.69281029452342728645500286642, 6.85474752694908283268138585662, 8.444229875574245306324191507419, 9.503747444756577207043289151852, 10.36207742334417150092590135815, 11.04979508964224836814588688269, 12.03469624354723640978823067954

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