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Properties

Label 2-260-13.3-c1-0-1
Degree $2$
Conductor $260$
Sign $0.0128 - 0.999i$
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  + (1.5 + 2.59i)3-s + 5-s + (−1.5 + 2.59i)7-s + (−3 + 5.19i)9-s + (−1.5 − 2.59i)11-s + (1 − 3.46i)13-s + (1.5 + 2.59i)15-s + (3.5 − 6.06i)17-s + (−0.5 + 0.866i)19-s − 9·21-s + (3.5 + 6.06i)23-s + 25-s − 9·27-s + (2.5 + 4.33i)29-s − 4·31-s + ⋯
L(s)  = 1  + (0.866 + 1.49i)3-s + 0.447·5-s + (−0.566 + 0.981i)7-s + (−1 + 1.73i)9-s + (−0.452 − 0.783i)11-s + (0.277 − 0.960i)13-s + (0.387 + 0.670i)15-s + (0.848 − 1.47i)17-s + (−0.114 + 0.198i)19-s − 1.96·21-s + (0.729 + 1.26i)23-s + 0.200·25-s − 1.73·27-s + (0.464 + 0.804i)29-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.0128 - 0.999i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.0128 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15265 + 1.13797i\)
\(L(\frac12)\) \(\approx\) \(1.15265 + 1.13797i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 + (-1 + 3.46i)T \)
good3 \( 1 + (-1.5 - 2.59i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (1.5 - 2.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.5 + 6.06i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.5 - 6.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.5 - 4.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-1.5 - 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.5 + 6.06i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.5 + 7.79i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (2.5 - 4.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.5 + 2.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 14T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (3.5 + 6.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.5 + 9.52i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.19853878755425947420649886630, −10.92717939296625100326435103603, −10.18152503572189814322031029344, −9.245426829227925592022661067875, −8.827687307576528227955970364243, −7.59608727551051792063104533307, −5.68511246731587629557269549968, −5.16667074973538874784221686285, −3.38456394538072386425508820427, −2.82742163210141865177049829200, 1.38598263478942632454871572922, 2.68229165966164450814091997592, 4.15126644541010437766477122150, 6.15033916662252742637679164847, 6.88566194168091777047892779835, 7.70986097863580839143133130907, 8.668143891591581185605195776330, 9.741117966510711587374384476735, 10.73318432269809254452033174415, 12.17554128371401982522629540324

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