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Properties

Label 2-260-65.63-c1-0-3
Degree $2$
Conductor $260$
Sign $0.931 + 0.362i$
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  + (0.387 − 1.44i)3-s + (1.95 + 1.07i)5-s + (2.10 + 1.21i)7-s + (0.655 + 0.378i)9-s + (−2.60 − 0.697i)11-s + (−3.57 + 0.447i)13-s + (2.32 − 2.41i)15-s + (6.05 − 1.62i)17-s + (−1.00 − 3.73i)19-s + (2.57 − 2.57i)21-s + (−1.14 − 0.307i)23-s + (2.66 + 4.22i)25-s + (3.97 − 3.97i)27-s + (−5.29 + 3.05i)29-s + (−6.22 − 6.22i)31-s + ⋯
L(s)  = 1  + (0.223 − 0.835i)3-s + (0.875 + 0.482i)5-s + (0.795 + 0.459i)7-s + (0.218 + 0.126i)9-s + (−0.784 − 0.210i)11-s + (−0.992 + 0.123i)13-s + (0.599 − 0.623i)15-s + (1.46 − 0.393i)17-s + (−0.229 − 0.856i)19-s + (0.561 − 0.561i)21-s + (−0.239 − 0.0641i)23-s + (0.533 + 0.845i)25-s + (0.765 − 0.765i)27-s + (−0.982 + 0.567i)29-s + (−1.11 − 1.11i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.362i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.931 + 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.931 + 0.362i$
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: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.931 + 0.362i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56727 - 0.294179i\)
\(L(\frac12)\) \(\approx\) \(1.56727 - 0.294179i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.95 - 1.07i)T \)
13 \( 1 + (3.57 - 0.447i)T \)
good3 \( 1 + (-0.387 + 1.44i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-2.10 - 1.21i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.60 + 0.697i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-6.05 + 1.62i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.00 + 3.73i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.14 + 0.307i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (5.29 - 3.05i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.22 + 6.22i)T + 31iT^{2} \)
37 \( 1 + (6.69 - 3.86i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.913 + 3.41i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.17 - 4.37i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 - 6.81iT - 47T^{2} \)
53 \( 1 + (-4.88 - 4.88i)T + 53iT^{2} \)
59 \( 1 + (9.04 - 2.42i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (4.87 - 8.44i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.507 - 0.878i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-10.5 + 2.83i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + 9.06T + 73T^{2} \)
79 \( 1 + 11.6iT - 79T^{2} \)
83 \( 1 + 1.09iT - 83T^{2} \)
89 \( 1 + (1.29 - 4.83i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.38 + 2.39i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.08836634278694525795871040003, −10.96835981745899442933664266514, −10.05212326544586767647998049030, −9.047386544583550924278951131169, −7.71841696020536715772953321453, −7.25029708833230923888659139367, −5.83097687800227179294948503910, −4.95044340292996818583122812149, −2.80447260267474674817014424155, −1.79182921428007821385987825969, 1.80173906219895269586188059284, 3.63150974063753609268756260503, 4.90198492287488349305809851516, 5.57633533300841249795820681414, 7.27995328052697703145938323160, 8.225513505197529525001581658894, 9.423547071191241314361784537180, 10.13529181187612484250282985950, 10.67647434632718517652102125690, 12.24470376531564440362181278269

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