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Properties

Label 2-260-20.7-c1-0-24
Degree $2$
Conductor $260$
Sign $-0.114 + 0.993i$
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  + (−0.942 + 1.05i)2-s + (0.158 + 0.158i)3-s + (−0.222 − 1.98i)4-s + (−2.23 + 0.0111i)5-s + (−0.315 + 0.0176i)6-s + (−2.20 + 2.20i)7-s + (2.30 + 1.63i)8-s − 2.94i·9-s + (2.09 − 2.36i)10-s − 3.71i·11-s + (0.279 − 0.349i)12-s + (0.707 − 0.707i)13-s + (−0.246 − 4.40i)14-s + (−0.355 − 0.352i)15-s + (−3.90 + 0.885i)16-s + (−3.39 − 3.39i)17-s + ⋯
L(s)  = 1  + (−0.666 + 0.745i)2-s + (0.0913 + 0.0913i)3-s + (−0.111 − 0.993i)4-s + (−0.999 + 0.00496i)5-s + (−0.128 + 0.00720i)6-s + (−0.834 + 0.834i)7-s + (0.815 + 0.579i)8-s − 0.983i·9-s + (0.662 − 0.748i)10-s − 1.12i·11-s + (0.0805 − 0.100i)12-s + (0.196 − 0.196i)13-s + (−0.0658 − 1.17i)14-s + (−0.0917 − 0.0908i)15-s + (−0.975 + 0.221i)16-s + (−0.824 − 0.824i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.114 + 0.993i$
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: \((2,\ 260,\ (\ :1/2),\ -0.114 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.186288 - 0.209088i\)
\(L(\frac12)\) \(\approx\) \(0.186288 - 0.209088i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.942 - 1.05i)T \)
5 \( 1 + (2.23 - 0.0111i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-0.158 - 0.158i)T + 3iT^{2} \)
7 \( 1 + (2.20 - 2.20i)T - 7iT^{2} \)
11 \( 1 + 3.71iT - 11T^{2} \)
17 \( 1 + (3.39 + 3.39i)T + 17iT^{2} \)
19 \( 1 + 3.95T + 19T^{2} \)
23 \( 1 + (5.15 + 5.15i)T + 23iT^{2} \)
29 \( 1 - 2.61iT - 29T^{2} \)
31 \( 1 + 1.82iT - 31T^{2} \)
37 \( 1 + (-7.13 - 7.13i)T + 37iT^{2} \)
41 \( 1 + 3.88T + 41T^{2} \)
43 \( 1 + (-0.0375 - 0.0375i)T + 43iT^{2} \)
47 \( 1 + (6.34 - 6.34i)T - 47iT^{2} \)
53 \( 1 + (0.298 - 0.298i)T - 53iT^{2} \)
59 \( 1 + 1.09T + 59T^{2} \)
61 \( 1 + 8.96T + 61T^{2} \)
67 \( 1 + (7.47 - 7.47i)T - 67iT^{2} \)
71 \( 1 + 12.5iT - 71T^{2} \)
73 \( 1 + (-6.19 + 6.19i)T - 73iT^{2} \)
79 \( 1 - 16.4T + 79T^{2} \)
83 \( 1 + (-1.62 - 1.62i)T + 83iT^{2} \)
89 \( 1 + 1.23iT - 89T^{2} \)
97 \( 1 + (11.5 + 11.5i)T + 97iT^{2} \)
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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

−11.62316078932383644122187413035, −10.70895145678292567392228129454, −9.471270842678740656089924730717, −8.751655675268378097046377514724, −8.047503588499407596506834909286, −6.58149382117121375390997913077, −6.12302377967305866742129700748, −4.48725355864919210395565973749, −3.00791708502313444973135605629, −0.25806379247481077795465855472, 2.05445950989555096112831059248, 3.73051449483052082639596576186, 4.45692537876909980440227031967, 6.70624378488204459960511111181, 7.58407106595484625630350931225, 8.310768654992478428609003728849, 9.546201936700090544237389024700, 10.45118145160531705457746543001, 11.09896589306287843077660949466, 12.17461331843908953543609228343

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