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Properties

Label 2-260-65.63-c1-0-6
Degree $2$
Conductor $260$
Sign $-0.541 + 0.840i$
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  + (0.877 − 3.27i)3-s + (−1.34 − 1.78i)5-s + (2.27 + 1.31i)7-s + (−7.34 − 4.24i)9-s + (1.71 + 0.458i)11-s + (1.21 + 3.39i)13-s + (−7.02 + 2.83i)15-s + (0.363 − 0.0972i)17-s + (−0.462 − 1.72i)19-s + (6.28 − 6.28i)21-s + (−4.72 − 1.26i)23-s + (−1.37 + 4.80i)25-s + (−13.1 + 13.1i)27-s + (6.28 − 3.62i)29-s + (2.98 + 2.98i)31-s + ⋯
L(s)  = 1  + (0.506 − 1.88i)3-s + (−0.601 − 0.798i)5-s + (0.859 + 0.495i)7-s + (−2.44 − 1.41i)9-s + (0.516 + 0.138i)11-s + (0.337 + 0.941i)13-s + (−1.81 + 0.733i)15-s + (0.0880 − 0.0235i)17-s + (−0.106 − 0.395i)19-s + (1.37 − 1.37i)21-s + (−0.986 − 0.264i)23-s + (−0.275 + 0.961i)25-s + (−2.52 + 2.52i)27-s + (1.16 − 0.674i)29-s + (0.535 + 0.535i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.541 + 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.541 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.541 + 0.840i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.541 + 0.840i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.661248 - 1.21182i\)
\(L(\frac12)\) \(\approx\) \(0.661248 - 1.21182i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.34 + 1.78i)T \)
13 \( 1 + (-1.21 - 3.39i)T \)
good3 \( 1 + (-0.877 + 3.27i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-2.27 - 1.31i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.71 - 0.458i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.363 + 0.0972i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.462 + 1.72i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (4.72 + 1.26i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-6.28 + 3.62i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.98 - 2.98i)T + 31iT^{2} \)
37 \( 1 + (-8.83 + 5.10i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.59 + 5.96i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.427 - 1.59i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 - 1.19iT - 47T^{2} \)
53 \( 1 + (-5.13 - 5.13i)T + 53iT^{2} \)
59 \( 1 + (2.89 - 0.775i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.45 - 4.25i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.98 - 6.90i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.02 + 1.07i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 - 2.72iT - 79T^{2} \)
83 \( 1 - 11.2iT - 83T^{2} \)
89 \( 1 + (-1.91 + 7.14i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.838 - 1.45i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.95603457463515864499726267291, −11.31928483604537428293331116732, −9.201755537394113503202091611987, −8.492638382806025015774757243797, −7.87428470475992422503686509636, −6.85636912577261053640288330568, −5.81887788784466110861123449425, −4.23721489180099167759956588890, −2.37482568825661604654850298266, −1.17034504722379731207719366866, 2.93702443866110235669384362549, 3.91053626658545818688757847837, 4.71807111067863175364550241836, 6.07768899987870816209216516663, 7.909301634985460913767098744886, 8.342366245180304572934634222599, 9.717345084654776543177826856738, 10.41330382110284034006379832575, 11.08148289557199637172627651028, 11.81901451768881582365069181454

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