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Properties

Label 2-260-65.63-c1-0-4
Degree $2$
Conductor $260$
Sign $-0.863 + 0.503i$
Analytic cond. $2.07611$
Root an. cond. $1.44087$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 1.86i)3-s + (−2 − i)5-s + (−3.86 − 2.23i)7-s + (−0.633 − 0.366i)9-s + (−2.86 − 0.767i)11-s + (−2 + 3i)13-s + (2.86 − 3.23i)15-s + (3.23 − 0.866i)17-s + (−1.13 − 4.23i)19-s + (6.09 − 6.09i)21-s + (−6.96 − 1.86i)23-s + (3 + 4i)25-s + (−3.09 + 3.09i)27-s + (−1.5 + 0.866i)29-s + (5.19 + 5.19i)31-s + ⋯
L(s)  = 1  + (−0.288 + 1.07i)3-s + (−0.894 − 0.447i)5-s + (−1.46 − 0.843i)7-s + (−0.211 − 0.122i)9-s + (−0.864 − 0.231i)11-s + (−0.554 + 0.832i)13-s + (0.740 − 0.834i)15-s + (0.783 − 0.210i)17-s + (−0.260 − 0.970i)19-s + (1.33 − 1.33i)21-s + (−1.45 − 0.389i)23-s + (0.600 + 0.800i)25-s + (−0.596 + 0.596i)27-s + (−0.278 + 0.160i)29-s + (0.933 + 0.933i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.863 + 0.503i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.863 + 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.863 + 0.503i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.863 + 0.503i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + i)T \)
13 \( 1 + (2 - 3i)T \)
good3 \( 1 + (0.5 - 1.86i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (3.86 + 2.23i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.86 + 0.767i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-3.23 + 0.866i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.13 + 4.23i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (6.96 + 1.86i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (1.5 - 0.866i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.19 - 5.19i)T + 31iT^{2} \)
37 \( 1 + (7.33 - 4.23i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.669 + 2.5i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.964 - 3.59i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + 2.92iT - 47T^{2} \)
53 \( 1 + (4.46 + 4.46i)T + 53iT^{2} \)
59 \( 1 + (6.33 - 1.69i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-4.5 + 7.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.86 - 13.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.59 - 1.23i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + 1.07T + 73T^{2} \)
79 \( 1 + 7.46iT - 79T^{2} \)
83 \( 1 + 10.9iT - 83T^{2} \)
89 \( 1 + (-0.794 + 2.96i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.13 + 3.69i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.52245413328551044038987535151, −10.37198160571133319966520668730, −9.960248250572007435468962133060, −8.896618256684059699562887867796, −7.61496261900778183029399949108, −6.63928451917899978981395186298, −5.09661681544614969419663511883, −4.20406615954836237219550413297, −3.23350443482501049199921909456, 0, 2.47883410443463153285438854720, 3.68595093504021799051538127202, 5.66105279659693649096274484464, 6.40311872030447176096305503975, 7.56850270455174738563495620025, 8.067338261541585883227190416075, 9.707462697741173191870162649952, 10.41192125148221359957302811287, 11.90494379124894383378596945149

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