Properties

Label 2-260-20.7-c1-0-13
Degree $2$
Conductor $260$
Sign $0.879 + 0.475i$
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  + (−1.32 − 0.487i)2-s + (0.396 + 0.396i)3-s + (1.52 + 1.29i)4-s + (2.02 − 0.951i)5-s + (−0.333 − 0.720i)6-s + (0.477 − 0.477i)7-s + (−1.39 − 2.46i)8-s − 2.68i·9-s + (−3.15 + 0.277i)10-s + 4.12i·11-s + (0.0920 + 1.11i)12-s + (0.707 − 0.707i)13-s + (−0.865 + 0.401i)14-s + (1.18 + 0.425i)15-s + (0.653 + 3.94i)16-s + (−0.282 − 0.282i)17-s + ⋯
L(s)  = 1  + (−0.938 − 0.344i)2-s + (0.229 + 0.229i)3-s + (0.762 + 0.646i)4-s + (0.904 − 0.425i)5-s + (−0.136 − 0.293i)6-s + (0.180 − 0.180i)7-s + (−0.493 − 0.869i)8-s − 0.895i·9-s + (−0.996 + 0.0877i)10-s + 1.24i·11-s + (0.0265 + 0.322i)12-s + (0.196 − 0.196i)13-s + (−0.231 + 0.107i)14-s + (0.304 + 0.109i)15-s + (0.163 + 0.986i)16-s + (−0.0685 − 0.0685i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.879 + 0.475i$
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: \((2,\ 260,\ (\ :1/2),\ 0.879 + 0.475i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03935 - 0.262981i\)
\(L(\frac12)\) \(\approx\) \(1.03935 - 0.262981i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 + 0.487i)T \)
5 \( 1 + (-2.02 + 0.951i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-0.396 - 0.396i)T + 3iT^{2} \)
7 \( 1 + (-0.477 + 0.477i)T - 7iT^{2} \)
11 \( 1 - 4.12iT - 11T^{2} \)
17 \( 1 + (0.282 + 0.282i)T + 17iT^{2} \)
19 \( 1 - 6.79T + 19T^{2} \)
23 \( 1 + (4.18 + 4.18i)T + 23iT^{2} \)
29 \( 1 + 2.26iT - 29T^{2} \)
31 \( 1 - 6.38iT - 31T^{2} \)
37 \( 1 + (-3.49 - 3.49i)T + 37iT^{2} \)
41 \( 1 - 3.13T + 41T^{2} \)
43 \( 1 + (3.14 + 3.14i)T + 43iT^{2} \)
47 \( 1 + (2.10 - 2.10i)T - 47iT^{2} \)
53 \( 1 + (3.55 - 3.55i)T - 53iT^{2} \)
59 \( 1 + 11.1T + 59T^{2} \)
61 \( 1 - 1.24T + 61T^{2} \)
67 \( 1 + (8.43 - 8.43i)T - 67iT^{2} \)
71 \( 1 + 14.8iT - 71T^{2} \)
73 \( 1 + (3.39 - 3.39i)T - 73iT^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 + (-0.412 - 0.412i)T + 83iT^{2} \)
89 \( 1 - 16.9iT - 89T^{2} \)
97 \( 1 + (2.59 + 2.59i)T + 97iT^{2} \)
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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

−12.02183498593856491921789157217, −10.63532593979738725237825157939, −9.756366458245040325311547377938, −9.356140947311728598444264846279, −8.259541605499303622175648826630, −7.12883830890816778441300459054, −6.06305160613233910973823273041, −4.47664098491548314578664405285, −2.90299823270037892202226704283, −1.38604682891880942599804176383, 1.65174822970420910460536596641, 2.97450702457495152303342134628, 5.39230722563172931758504175585, 6.07563306255237920338817323239, 7.35185505740514364983977482459, 8.149540351262292926703394751007, 9.193075497377490376733316488652, 9.995716063731118104100301759081, 11.01307566099987820170712747167, 11.61191804595689228964817626365

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