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Properties

Label 2-260-20.3-c1-0-11
Degree $2$
Conductor $260$
Sign $0.514 - 0.857i$
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  + (1.38 − 0.293i)2-s + (−1.30 + 1.30i)3-s + (1.82 − 0.811i)4-s + (−2.07 + 0.831i)5-s + (−1.41 + 2.18i)6-s + (2.44 + 2.44i)7-s + (2.29 − 1.65i)8-s − 0.383i·9-s + (−2.62 + 1.75i)10-s + 4.04i·11-s + (−1.32 + 3.43i)12-s + (−0.707 − 0.707i)13-s + (4.10 + 2.66i)14-s + (1.61 − 3.78i)15-s + (2.68 − 2.96i)16-s + (5.07 − 5.07i)17-s + ⋯
L(s)  = 1  + (0.978 − 0.207i)2-s + (−0.750 + 0.750i)3-s + (0.913 − 0.405i)4-s + (−0.928 + 0.372i)5-s + (−0.578 + 0.890i)6-s + (0.925 + 0.925i)7-s + (0.809 − 0.586i)8-s − 0.127i·9-s + (−0.830 + 0.556i)10-s + 1.21i·11-s + (−0.381 + 0.991i)12-s + (−0.196 − 0.196i)13-s + (1.09 + 0.713i)14-s + (0.417 − 0.976i)15-s + (0.670 − 0.741i)16-s + (1.23 − 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.514 - 0.857i$
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: \((2,\ 260,\ (\ :1/2),\ 0.514 - 0.857i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46137 + 0.827878i\)
\(L(\frac12)\) \(\approx\) \(1.46137 + 0.827878i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 + 0.293i)T \)
5 \( 1 + (2.07 - 0.831i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (1.30 - 1.30i)T - 3iT^{2} \)
7 \( 1 + (-2.44 - 2.44i)T + 7iT^{2} \)
11 \( 1 - 4.04iT - 11T^{2} \)
17 \( 1 + (-5.07 + 5.07i)T - 17iT^{2} \)
19 \( 1 + 3.28T + 19T^{2} \)
23 \( 1 + (1.98 - 1.98i)T - 23iT^{2} \)
29 \( 1 - 5.17iT - 29T^{2} \)
31 \( 1 + 5.36iT - 31T^{2} \)
37 \( 1 + (-5.53 + 5.53i)T - 37iT^{2} \)
41 \( 1 + 9.24T + 41T^{2} \)
43 \( 1 + (-6.42 + 6.42i)T - 43iT^{2} \)
47 \( 1 + (-1.30 - 1.30i)T + 47iT^{2} \)
53 \( 1 + (3.32 + 3.32i)T + 53iT^{2} \)
59 \( 1 - 4.58T + 59T^{2} \)
61 \( 1 - 6.40T + 61T^{2} \)
67 \( 1 + (3.98 + 3.98i)T + 67iT^{2} \)
71 \( 1 - 8.66iT - 71T^{2} \)
73 \( 1 + (-8.05 - 8.05i)T + 73iT^{2} \)
79 \( 1 - 1.43T + 79T^{2} \)
83 \( 1 + (5.26 - 5.26i)T - 83iT^{2} \)
89 \( 1 + 8.95iT - 89T^{2} \)
97 \( 1 + (-0.869 + 0.869i)T - 97iT^{2} \)
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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

−11.91549603898996836866935096176, −11.48078928458615857310592678978, −10.59070301410703769923446812268, −9.681780559364513534771701354395, −7.967956286608563790977099568656, −7.09908303866150927244678713179, −5.55710680789398449821916227677, −4.93574502547165174310671158563, −3.98196966321695198477474551636, −2.36855036541756363039536906815, 1.22772979361424656605635614740, 3.56511259113644279521187385679, 4.53772243849666052155033443927, 5.77068928846248507342948370310, 6.67640362273469695990783884675, 7.82100769399377236052781537640, 8.277245825815604160861081618542, 10.53796200696067115983741473487, 11.25798562315871874279682153567, 11.94841150529669248155940678375

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