Properties

Label 2-260-13.12-c1-0-2
Degree $2$
Conductor $260$
Sign $-0.349 + 0.936i$
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  − 2.26·3-s + i·5-s − 1.11i·7-s + 2.11·9-s − 5.37i·11-s + (−3.37 − 1.26i)13-s − 2.26i·15-s − 7.90i·19-s + 2.52i·21-s − 6.49·23-s − 25-s + 2·27-s + 3.63·29-s + 3.14i·31-s + 12.1i·33-s + ⋯
L(s)  = 1  − 1.30·3-s + 0.447i·5-s − 0.421i·7-s + 0.705·9-s − 1.62i·11-s + (−0.936 − 0.349i)13-s − 0.583i·15-s − 1.81i·19-s + 0.550i·21-s − 1.35·23-s − 0.200·25-s + 0.384·27-s + 0.675·29-s + 0.565i·31-s + 2.11i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.349 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.349 + 0.936i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.349 + 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.282042 - 0.406448i\)
\(L(\frac12)\) \(\approx\) \(0.282042 - 0.406448i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - iT \)
13 \( 1 + (3.37 + 1.26i)T \)
good3 \( 1 + 2.26T + 3T^{2} \)
7 \( 1 + 1.11iT - 7T^{2} \)
11 \( 1 + 5.37iT - 11T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 7.90iT - 19T^{2} \)
23 \( 1 + 6.49T + 23T^{2} \)
29 \( 1 - 3.63T + 29T^{2} \)
31 \( 1 - 3.14iT - 31T^{2} \)
37 \( 1 + 7.40iT - 37T^{2} \)
41 \( 1 - 4.75iT - 41T^{2} \)
43 \( 1 + 4.78T + 43T^{2} \)
47 \( 1 - 6.16iT - 47T^{2} \)
53 \( 1 - 0.292T + 53T^{2} \)
59 \( 1 - 11.3iT - 59T^{2} \)
61 \( 1 + 5.34T + 61T^{2} \)
67 \( 1 + 11.8iT - 67T^{2} \)
71 \( 1 + 3.43iT - 71T^{2} \)
73 \( 1 + 4.59iT - 73T^{2} \)
79 \( 1 + 10.8T + 79T^{2} \)
83 \( 1 - 7.40iT - 83T^{2} \)
89 \( 1 + 14.8iT - 89T^{2} \)
97 \( 1 - 3.76iT - 97T^{2} \)
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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

−11.51467276811020185216378042810, −10.89749419927772390565708541732, −10.17450068876300009305252540840, −8.833709285733065172238730399308, −7.54311122789817694711036219218, −6.49953412513349340777205932718, −5.69118701232763809820752689604, −4.58746939953694681042598312506, −2.95807887502433541624028549499, −0.44046045040546209234330452445, 1.93901757859312782630584221894, 4.25599268122250891520053805983, 5.17422814019535993076890703684, 6.10694181351273231024228946637, 7.19554952186384732275227899495, 8.336880769128702584993946747788, 9.878601338544687943855587556675, 10.15271561514571770410132780729, 11.74011611367768377829511017224, 12.10977350944044636678888220779

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