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Properties

Label 2-260-65.58-c1-0-6
Degree $2$
Conductor $260$
Sign $0.917 + 0.398i$
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  + (2.57 − 0.690i)3-s + (2.23 + 0.0143i)5-s + (−1.08 − 1.87i)7-s + (3.56 − 2.06i)9-s + (−5.13 + 1.37i)11-s + (−2.72 + 2.35i)13-s + (5.77 − 1.50i)15-s + (0.288 − 1.07i)17-s + (−1.69 + 6.33i)19-s + (−4.08 − 4.08i)21-s + (−0.192 − 0.718i)23-s + (4.99 + 0.0642i)25-s + (2.11 − 2.11i)27-s + (0.0866 + 0.0500i)29-s + (3.90 − 3.90i)31-s + ⋯
L(s)  = 1  + (1.48 − 0.398i)3-s + (0.999 + 0.00642i)5-s + (−0.408 − 0.708i)7-s + (1.18 − 0.686i)9-s + (−1.54 + 0.414i)11-s + (−0.757 + 0.653i)13-s + (1.49 − 0.389i)15-s + (0.0699 − 0.260i)17-s + (−0.389 + 1.45i)19-s + (−0.890 − 0.890i)21-s + (−0.0401 − 0.149i)23-s + (0.999 + 0.0128i)25-s + (0.406 − 0.406i)27-s + (0.0160 + 0.00929i)29-s + (0.700 − 0.700i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.917 + 0.398i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.917 + 0.398i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95986 - 0.407703i\)
\(L(\frac12)\) \(\approx\) \(1.95986 - 0.407703i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 - 0.0143i)T \)
13 \( 1 + (2.72 - 2.35i)T \)
good3 \( 1 + (-2.57 + 0.690i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (1.08 + 1.87i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (5.13 - 1.37i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.288 + 1.07i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.69 - 6.33i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.192 + 0.718i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-0.0866 - 0.0500i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.90 + 3.90i)T - 31iT^{2} \)
37 \( 1 + (-3.41 + 5.91i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.36 - 5.10i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-3.57 - 0.959i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 2.04T + 47T^{2} \)
53 \( 1 + (-8.28 - 8.28i)T + 53iT^{2} \)
59 \( 1 + (12.1 + 3.25i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (4.66 + 8.08i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.54 + 5.51i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (9.19 + 2.46i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 - 10.7iT - 73T^{2} \)
79 \( 1 + 15.3iT - 79T^{2} \)
83 \( 1 - 0.473T + 83T^{2} \)
89 \( 1 + (-0.560 - 2.09i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-10.9 + 6.34i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.37383283230080853794129875190, −10.54581957629565257193493027215, −9.898964048803078691112068024868, −9.128268014998640087002368649631, −7.896738088320110446607027698599, −7.31904509461510067414762456846, −5.99046711432955020574987307254, −4.44676230054554322044166428378, −2.91397709444772044590990889825, −2.00012914493419938429366168946, 2.54340330828619056062641597553, 2.87291599146952085741073903089, 4.80534764175360936488619663203, 5.85851583118023189797611113619, 7.36125638987316305366170685829, 8.452828391660039298287564338038, 9.096645162915552187894839782794, 10.00907293048234709215844620813, 10.65755843512845852794070423231, 12.38389932000541900712089273676

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