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Properties

Label 2-260-20.7-c1-0-12
Degree $2$
Conductor $260$
Sign $-0.399 - 0.916i$
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  + (0.915 + 1.07i)2-s + (1.47 + 1.47i)3-s + (−0.323 + 1.97i)4-s + (−1.55 + 1.61i)5-s + (−0.239 + 2.94i)6-s + (2.13 − 2.13i)7-s + (−2.42 + 1.45i)8-s + 1.36i·9-s + (−3.15 − 0.195i)10-s − 4.29i·11-s + (−3.39 + 2.43i)12-s + (0.707 − 0.707i)13-s + (4.25 + 0.346i)14-s + (−4.67 + 0.0906i)15-s + (−3.79 − 1.27i)16-s + (1.46 + 1.46i)17-s + ⋯
L(s)  = 1  + (0.647 + 0.762i)2-s + (0.853 + 0.853i)3-s + (−0.161 + 0.986i)4-s + (−0.693 + 0.720i)5-s + (−0.0979 + 1.20i)6-s + (0.807 − 0.807i)7-s + (−0.856 + 0.515i)8-s + 0.456i·9-s + (−0.998 − 0.0617i)10-s − 1.29i·11-s + (−0.980 + 0.704i)12-s + (0.196 − 0.196i)13-s + (1.13 + 0.0926i)14-s + (−1.20 + 0.0234i)15-s + (−0.947 − 0.319i)16-s + (0.355 + 0.355i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.399 - 0.916i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.399 - 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12194 + 1.71199i\)
\(L(\frac12)\) \(\approx\) \(1.12194 + 1.71199i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.915 - 1.07i)T \)
5 \( 1 + (1.55 - 1.61i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-1.47 - 1.47i)T + 3iT^{2} \)
7 \( 1 + (-2.13 + 2.13i)T - 7iT^{2} \)
11 \( 1 + 4.29iT - 11T^{2} \)
17 \( 1 + (-1.46 - 1.46i)T + 17iT^{2} \)
19 \( 1 + 3.97T + 19T^{2} \)
23 \( 1 + (-4.23 - 4.23i)T + 23iT^{2} \)
29 \( 1 - 4.88iT - 29T^{2} \)
31 \( 1 - 2.60iT - 31T^{2} \)
37 \( 1 + (2.66 + 2.66i)T + 37iT^{2} \)
41 \( 1 - 8.74T + 41T^{2} \)
43 \( 1 + (8.10 + 8.10i)T + 43iT^{2} \)
47 \( 1 + (4.30 - 4.30i)T - 47iT^{2} \)
53 \( 1 + (3.75 - 3.75i)T - 53iT^{2} \)
59 \( 1 + 8.15T + 59T^{2} \)
61 \( 1 + 3.28T + 61T^{2} \)
67 \( 1 + (-5.76 + 5.76i)T - 67iT^{2} \)
71 \( 1 + 11.6iT - 71T^{2} \)
73 \( 1 + (-7.95 + 7.95i)T - 73iT^{2} \)
79 \( 1 - 0.366T + 79T^{2} \)
83 \( 1 + (6.72 + 6.72i)T + 83iT^{2} \)
89 \( 1 - 13.0iT - 89T^{2} \)
97 \( 1 + (4.31 + 4.31i)T + 97iT^{2} \)
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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

−12.41744691042153982987257185431, −11.12339049200802268847609947540, −10.66362425554858940806998508275, −9.047526385128293598874887461760, −8.256243492652300606502663115590, −7.51692525563515367586495247674, −6.31830337602610990974927838518, −4.85374256891764629994049647778, −3.74897369641316219700404901121, −3.19644355179656582057174485003, 1.59810806898679879057977560888, 2.63064600197756041444682021505, 4.31227684167279692762582325382, 5.13239585856268705733772763793, 6.74719523614770397670128101887, 7.978777184950302538930205351960, 8.702665298695811357140261857036, 9.700473139940320478434129915043, 11.14357222429258688920952024307, 11.93759972557757109429965004603

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