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Properties

Label 2-260-20.7-c1-0-8
Degree $2$
Conductor $260$
Sign $0.827 - 0.561i$
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  + (−0.769 + 1.18i)2-s + (−0.808 − 0.808i)3-s + (−0.817 − 1.82i)4-s + (0.885 + 2.05i)5-s + (1.58 − 0.337i)6-s + (0.771 − 0.771i)7-s + (2.79 + 0.433i)8-s − 1.69i·9-s + (−3.11 − 0.527i)10-s − 0.875i·11-s + (−0.815 + 2.13i)12-s + (0.707 − 0.707i)13-s + (0.322 + 1.50i)14-s + (0.943 − 2.37i)15-s + (−2.66 + 2.98i)16-s + (4.91 + 4.91i)17-s + ⋯
L(s)  = 1  + (−0.543 + 0.839i)2-s + (−0.466 − 0.466i)3-s + (−0.408 − 0.912i)4-s + (0.396 + 0.918i)5-s + (0.645 − 0.137i)6-s + (0.291 − 0.291i)7-s + (0.988 + 0.153i)8-s − 0.563i·9-s + (−0.985 − 0.166i)10-s − 0.264i·11-s + (−0.235 + 0.616i)12-s + (0.196 − 0.196i)13-s + (0.0861 + 0.403i)14-s + (0.243 − 0.613i)15-s + (−0.666 + 0.745i)16-s + (1.19 + 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.827 - 0.561i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.827 - 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.882839 + 0.271397i\)
\(L(\frac12)\) \(\approx\) \(0.882839 + 0.271397i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.769 - 1.18i)T \)
5 \( 1 + (-0.885 - 2.05i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (0.808 + 0.808i)T + 3iT^{2} \)
7 \( 1 + (-0.771 + 0.771i)T - 7iT^{2} \)
11 \( 1 + 0.875iT - 11T^{2} \)
17 \( 1 + (-4.91 - 4.91i)T + 17iT^{2} \)
19 \( 1 - 6.89T + 19T^{2} \)
23 \( 1 + (-6.03 - 6.03i)T + 23iT^{2} \)
29 \( 1 + 7.85iT - 29T^{2} \)
31 \( 1 + 5.45iT - 31T^{2} \)
37 \( 1 + (2.21 + 2.21i)T + 37iT^{2} \)
41 \( 1 + 1.36T + 41T^{2} \)
43 \( 1 + (4.34 + 4.34i)T + 43iT^{2} \)
47 \( 1 + (-3.81 + 3.81i)T - 47iT^{2} \)
53 \( 1 + (2.18 - 2.18i)T - 53iT^{2} \)
59 \( 1 + 0.749T + 59T^{2} \)
61 \( 1 + 4.79T + 61T^{2} \)
67 \( 1 + (10.0 - 10.0i)T - 67iT^{2} \)
71 \( 1 + 5.01iT - 71T^{2} \)
73 \( 1 + (-6.85 + 6.85i)T - 73iT^{2} \)
79 \( 1 + 1.62T + 79T^{2} \)
83 \( 1 + (2.66 + 2.66i)T + 83iT^{2} \)
89 \( 1 - 3.68iT - 89T^{2} \)
97 \( 1 + (-6.65 - 6.65i)T + 97iT^{2} \)
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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

−11.86782289142791261605021415541, −11.01498165960975678089548601192, −10.05229636665206974852074323271, −9.267733260837690643400131937773, −7.79292328484808263699437398626, −7.23147523389067527755793582625, −6.07258981934103798856638522829, −5.52963726309446924311493843582, −3.54934793031314028018232239969, −1.27827706634727027198069554552, 1.30068546197446560560855618037, 3.03680219669892756700343137296, 4.83936066700335034108873595491, 5.20161728537240796363734111809, 7.21132601200113862113260700599, 8.337622416150002605294118838492, 9.244660251027148776761790868367, 9.987777256682275476264108405381, 10.91099590132179039448759844020, 11.85457649526911388928019629633

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