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Properties

Label 2-260-20.3-c1-0-0
Degree $2$
Conductor $260$
Sign $-0.494 + 0.869i$
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  + (0.0221 + 1.41i)2-s + (−0.798 + 0.798i)3-s + (−1.99 + 0.0625i)4-s + (−1.57 + 1.58i)5-s + (−1.14 − 1.11i)6-s + (−1.30 − 1.30i)7-s + (−0.132 − 2.82i)8-s + 1.72i·9-s + (−2.28 − 2.18i)10-s − 3.52i·11-s + (1.54 − 1.64i)12-s + (−0.707 − 0.707i)13-s + (1.81 − 1.87i)14-s + (−0.0124 − 2.52i)15-s + (3.99 − 0.250i)16-s + (−1.96 + 1.96i)17-s + ⋯
L(s)  = 1  + (0.0156 + 0.999i)2-s + (−0.461 + 0.461i)3-s + (−0.999 + 0.0312i)4-s + (−0.703 + 0.710i)5-s + (−0.468 − 0.453i)6-s + (−0.493 − 0.493i)7-s + (−0.0469 − 0.998i)8-s + 0.574i·9-s + (−0.721 − 0.692i)10-s − 1.06i·11-s + (0.446 − 0.475i)12-s + (−0.196 − 0.196i)13-s + (0.485 − 0.501i)14-s + (−0.00322 − 0.652i)15-s + (0.998 − 0.0625i)16-s + (−0.476 + 0.476i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.494 + 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.494 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.494 + 0.869i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.494 + 0.869i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.141388 - 0.243130i\)
\(L(\frac12)\) \(\approx\) \(0.141388 - 0.243130i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0221 - 1.41i)T \)
5 \( 1 + (1.57 - 1.58i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (0.798 - 0.798i)T - 3iT^{2} \)
7 \( 1 + (1.30 + 1.30i)T + 7iT^{2} \)
11 \( 1 + 3.52iT - 11T^{2} \)
17 \( 1 + (1.96 - 1.96i)T - 17iT^{2} \)
19 \( 1 + 2.70T + 19T^{2} \)
23 \( 1 + (4.89 - 4.89i)T - 23iT^{2} \)
29 \( 1 - 6.90iT - 29T^{2} \)
31 \( 1 + 4.15iT - 31T^{2} \)
37 \( 1 + (6.10 - 6.10i)T - 37iT^{2} \)
41 \( 1 - 8.91T + 41T^{2} \)
43 \( 1 + (7.58 - 7.58i)T - 43iT^{2} \)
47 \( 1 + (-5.48 - 5.48i)T + 47iT^{2} \)
53 \( 1 + (4.55 + 4.55i)T + 53iT^{2} \)
59 \( 1 - 2.28T + 59T^{2} \)
61 \( 1 + 7.76T + 61T^{2} \)
67 \( 1 + (-0.898 - 0.898i)T + 67iT^{2} \)
71 \( 1 - 14.2iT - 71T^{2} \)
73 \( 1 + (-3.84 - 3.84i)T + 73iT^{2} \)
79 \( 1 - 0.766T + 79T^{2} \)
83 \( 1 + (-0.727 + 0.727i)T - 83iT^{2} \)
89 \( 1 + 0.442iT - 89T^{2} \)
97 \( 1 + (-5.22 + 5.22i)T - 97iT^{2} \)
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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

−12.82737492844368560175525256973, −11.46476470312825977474021431590, −10.64637909619082124290913955355, −9.856849424285481359285472805316, −8.467631958145916129424131967895, −7.68537280657691265722067618450, −6.61933327546995550933561162293, −5.72323886807462938554704695603, −4.41142644946188226460732002445, −3.42271110765776348644600406171, 0.22575427220943796845993460206, 2.13787946236311104754227307498, 3.86169361801675867071138397747, 4.83615818998909039482877588727, 6.18292872617670330550577207396, 7.49129166329561743862518680318, 8.769434905060325090707118953897, 9.420850658341140814169673333206, 10.54122350238075893776672468295, 11.72268200743988424914952961508

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