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Properties

Label 2-260-13.3-c1-0-2
Degree $2$
Conductor $260$
Sign $0.522 + 0.852i$
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  + (−0.5 − 0.866i)3-s + 5-s + (0.5 − 0.866i)7-s + (1 − 1.73i)9-s + (−1.5 − 2.59i)11-s + (1 + 3.46i)13-s + (−0.5 − 0.866i)15-s + (1.5 − 2.59i)17-s + (3.5 − 6.06i)19-s − 0.999·21-s + (1.5 + 2.59i)23-s + 25-s − 5·27-s + (−1.5 − 2.59i)29-s − 4·31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + 0.447·5-s + (0.188 − 0.327i)7-s + (0.333 − 0.577i)9-s + (−0.452 − 0.783i)11-s + (0.277 + 0.960i)13-s + (−0.129 − 0.223i)15-s + (0.363 − 0.630i)17-s + (0.802 − 1.39i)19-s − 0.218·21-s + (0.312 + 0.541i)23-s + 0.200·25-s − 0.962·27-s + (−0.278 − 0.482i)29-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.522 + 0.852i$
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: \((2,\ 260,\ (\ :1/2),\ 0.522 + 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09579 - 0.614058i\)
\(L(\frac12)\) \(\approx\) \(1.09579 - 0.614058i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.5 - 9.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.5 - 9.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.5 + 2.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (7.5 + 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.5 + 6.06i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.57063720767422974062168494641, −11.25450457241074693523981802724, −9.775142104213168411996169639242, −9.121818417072777770932432991858, −7.73438706040159902965256644242, −6.83210411564571409047157481953, −5.88413821826063427914267247536, −4.62920393158057106026919346181, −3.04107735360159192429699392409, −1.17743233124008280959897587172, 1.97844682059782919753617350785, 3.71086553297806021306157772874, 5.15902425992894204162895801001, 5.73676857566534083124455674535, 7.33093914032845698701516969171, 8.210209327450871420345092967992, 9.505655445924025053132115558918, 10.35156453584152564180173308457, 10.87247590479960051581959150199, 12.31543963923187685107399222014

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