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Properties

Label 2-260-65.49-c1-0-2
Degree $2$
Conductor $260$
Sign $-0.268 - 0.963i$
Analytic cond. $2.07611$
Root an. cond. $1.44087$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.180 − 0.104i)3-s + (−1.60 + 1.56i)5-s + (−1.72 + 2.99i)7-s + (−1.47 + 2.56i)9-s + (0.625 − 0.360i)11-s + (−3.18 − 1.69i)13-s + (−0.126 + 0.449i)15-s + (3.10 + 1.79i)17-s + (6.51 + 3.76i)19-s + 0.721i·21-s + (−2.05 + 1.18i)23-s + (0.125 − 4.99i)25-s + 1.24i·27-s + (−3.68 − 6.37i)29-s + 0.668i·31-s + ⋯
L(s)  = 1  + (0.104 − 0.0602i)3-s + (−0.715 + 0.698i)5-s + (−0.653 + 1.13i)7-s + (−0.492 + 0.853i)9-s + (0.188 − 0.108i)11-s + (−0.883 − 0.468i)13-s + (−0.0326 + 0.116i)15-s + (0.754 + 0.435i)17-s + (1.49 + 0.862i)19-s + 0.157i·21-s + (−0.428 + 0.247i)23-s + (0.0250 − 0.999i)25-s + 0.239i·27-s + (−0.683 − 1.18i)29-s + 0.120i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.268 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.268 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.268 - 0.963i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.268 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.526709 + 0.693707i\)
\(L(\frac12)\) \(\approx\) \(0.526709 + 0.693707i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (1.60 - 1.56i)T \)
13 \( 1 + (3.18 + 1.69i)T \)
good3 \( 1 + (-0.180 + 0.104i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (1.72 - 2.99i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.625 + 0.360i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.10 - 1.79i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.51 - 3.76i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.05 - 1.18i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.68 + 6.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.668iT - 31T^{2} \)
37 \( 1 + (-3.36 - 5.82i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.32 - 3.65i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.06 - 4.07i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.40T + 47T^{2} \)
53 \( 1 + 11.7iT - 53T^{2} \)
59 \( 1 + (4.20 + 2.42i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.68 + 4.64i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.80 - 13.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.10 + 2.37i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.36T + 73T^{2} \)
79 \( 1 - 5.20T + 79T^{2} \)
83 \( 1 - 13.3T + 83T^{2} \)
89 \( 1 + (-4.20 + 2.42i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.83 + 3.17i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.99898106236352708240780626702, −11.61033154565786036338559731571, −10.28011611535563940854857178879, −9.532571109845945221743177274213, −8.124849029551954741910460535266, −7.61420808691340887021088733705, −6.16427239291705964814420109978, −5.26812374660175889559455365893, −3.50433129311212894518633398772, −2.52646602828818598245556756910, 0.67553306623477952213714326912, 3.23276321109981839718591749644, 4.19881617576222622611007641750, 5.46397020647065361927066936922, 7.04441238689809753653168659351, 7.55036723111637250455775933983, 9.088645276124410643463007307589, 9.537629865686597929012036064011, 10.80660048773557861576193931942, 11.95527757564817235353369122553

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