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Properties

Label 2-260-13.4-c1-0-2
Degree $2$
Conductor $260$
Sign $0.856 - 0.515i$
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  + (−0.800 + 1.38i)3-s − i·5-s + (3.75 − 2.16i)7-s + (0.219 + 0.380i)9-s + (1.5 + 0.866i)11-s + (−3.11 + 1.81i)13-s + (1.38 + 0.800i)15-s + (3.75 + 6.49i)17-s + (4.65 − 2.68i)19-s + 6.93i·21-s + (−0.580 + 1.00i)23-s − 25-s − 5.50·27-s + (1.01 − 1.75i)29-s − 7.86i·31-s + ⋯
L(s)  = 1  + (−0.461 + 0.800i)3-s − 0.447i·5-s + (1.41 − 0.818i)7-s + (0.0732 + 0.126i)9-s + (0.452 + 0.261i)11-s + (−0.863 + 0.504i)13-s + (0.357 + 0.206i)15-s + (0.909 + 1.57i)17-s + (1.06 − 0.616i)19-s + 1.51i·21-s + (−0.121 + 0.209i)23-s − 0.200·25-s − 1.05·27-s + (0.187 − 0.325i)29-s − 1.41i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.856 - 0.515i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.856 - 0.515i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22297 + 0.339725i\)
\(L(\frac12)\) \(\approx\) \(1.22297 + 0.339725i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.11 - 1.81i)T \)
good3 \( 1 + (0.800 - 1.38i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-3.75 + 2.16i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.5 - 0.866i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.75 - 6.49i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.65 + 2.68i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.580 - 1.00i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.01 + 1.75i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.86iT - 31T^{2} \)
37 \( 1 + (8.25 + 4.76i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.69 - 3.86i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.09 - 3.62i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 + 12.6T + 53T^{2} \)
59 \( 1 + (-5.49 + 3.17i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.85 + 3.20i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.55 + 2.63i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (10.8 - 6.25i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 5.23iT - 73T^{2} \)
79 \( 1 + 8.16T + 79T^{2} \)
83 \( 1 - 0.456iT - 83T^{2} \)
89 \( 1 + (11.4 + 6.63i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.43 - 1.40i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.78403443584097281829289959697, −11.14334714102627786369478517717, −10.20617567214207872827760330233, −9.439535576186151345447233139200, −8.060284052732901410640100640258, −7.35971228430253325973728279824, −5.66556952044906277306580885754, −4.69644018803483543420545100368, −4.03421503065901668692345532547, −1.61517614365116339326544243150, 1.39986556318128548056087364612, 3.00947218793149423274226479325, 4.97294123988410212046570306359, 5.75218331614347540317694535222, 7.12735531189307262137125887124, 7.71823335052398096655621167211, 8.946230427809921154683229676980, 10.07069151761804094447778347421, 11.29377016224982480029539199269, 12.05507265919353402136166020535

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