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Properties

Label 2-260-20.3-c1-0-27
Degree $2$
Conductor $260$
Sign $0.677 + 0.735i$
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  + (−1.18 + 0.774i)2-s + (2.05 − 2.05i)3-s + (0.800 − 1.83i)4-s + (1.34 − 1.78i)5-s + (−0.841 + 4.03i)6-s + (0.845 + 0.845i)7-s + (0.472 + 2.78i)8-s − 5.48i·9-s + (−0.211 + 3.15i)10-s + 4.19i·11-s + (−2.12 − 5.42i)12-s + (−0.707 − 0.707i)13-s + (−1.65 − 0.345i)14-s + (−0.901 − 6.44i)15-s + (−2.71 − 2.93i)16-s + (−0.206 + 0.206i)17-s + ⋯
L(s)  = 1  + (−0.836 + 0.547i)2-s + (1.18 − 1.18i)3-s + (0.400 − 0.916i)4-s + (0.602 − 0.798i)5-s + (−0.343 + 1.64i)6-s + (0.319 + 0.319i)7-s + (0.167 + 0.985i)8-s − 1.82i·9-s + (−0.0668 + 0.997i)10-s + 1.26i·11-s + (−0.613 − 1.56i)12-s + (−0.196 − 0.196i)13-s + (−0.442 − 0.0923i)14-s + (−0.232 − 1.66i)15-s + (−0.679 − 0.733i)16-s + (−0.0499 + 0.0499i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.677 + 0.735i$
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: \((2,\ 260,\ (\ :1/2),\ 0.677 + 0.735i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24186 - 0.544454i\)
\(L(\frac12)\) \(\approx\) \(1.24186 - 0.544454i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.18 - 0.774i)T \)
5 \( 1 + (-1.34 + 1.78i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-2.05 + 2.05i)T - 3iT^{2} \)
7 \( 1 + (-0.845 - 0.845i)T + 7iT^{2} \)
11 \( 1 - 4.19iT - 11T^{2} \)
17 \( 1 + (0.206 - 0.206i)T - 17iT^{2} \)
19 \( 1 + 4.84T + 19T^{2} \)
23 \( 1 + (-0.389 + 0.389i)T - 23iT^{2} \)
29 \( 1 - 6.55iT - 29T^{2} \)
31 \( 1 - 5.06iT - 31T^{2} \)
37 \( 1 + (-8.49 + 8.49i)T - 37iT^{2} \)
41 \( 1 + 5.61T + 41T^{2} \)
43 \( 1 + (3.25 - 3.25i)T - 43iT^{2} \)
47 \( 1 + (-8.91 - 8.91i)T + 47iT^{2} \)
53 \( 1 + (-0.0714 - 0.0714i)T + 53iT^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 + 3.57T + 61T^{2} \)
67 \( 1 + (6.95 + 6.95i)T + 67iT^{2} \)
71 \( 1 - 4.61iT - 71T^{2} \)
73 \( 1 + (-2.57 - 2.57i)T + 73iT^{2} \)
79 \( 1 - 2.19T + 79T^{2} \)
83 \( 1 + (-11.1 + 11.1i)T - 83iT^{2} \)
89 \( 1 - 15.0iT - 89T^{2} \)
97 \( 1 + (5.51 - 5.51i)T - 97iT^{2} \)
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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

−12.26526335771945314569916956975, −10.60792027796527485093444007985, −9.416072331899215513773520737807, −8.865220555353824039151006053897, −8.030341846394077544850694716319, −7.20393790862123756947945871050, −6.21690168681720357682758019704, −4.81789143000140728358159969750, −2.37950612741690324092583619719, −1.51448678024610146727083675230, 2.29401474144546655903742279797, 3.26173678437773459068906604756, 4.30977302605483677265833072611, 6.24018856946977272650101411055, 7.70251309358910595196092159449, 8.522245900310702297841898570733, 9.350067841253985573126931240136, 10.16981841571746133781286045287, 10.76597495198543245498758888440, 11.60261404042065375134296782694

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