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Properties

Label 2-260-65.33-c1-0-0
Degree $2$
Conductor $260$
Sign $0.0308 - 0.999i$
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  + (−2.51 + 0.674i)3-s + (2.14 − 0.647i)5-s + (−1.45 + 0.839i)7-s + (3.29 − 1.90i)9-s + (0.758 + 2.83i)11-s + (0.727 + 3.53i)13-s + (−4.95 + 3.07i)15-s + (−2.11 + 7.90i)17-s + (−2.32 − 0.621i)19-s + (3.09 − 3.09i)21-s + (0.421 + 1.57i)23-s + (4.16 − 2.77i)25-s + (−1.47 + 1.47i)27-s + (−2.41 − 1.39i)29-s + (7.32 + 7.32i)31-s + ⋯
L(s)  = 1  + (−1.45 + 0.389i)3-s + (0.957 − 0.289i)5-s + (−0.549 + 0.317i)7-s + (1.09 − 0.633i)9-s + (0.228 + 0.853i)11-s + (0.201 + 0.979i)13-s + (−1.27 + 0.794i)15-s + (−0.513 + 1.91i)17-s + (−0.532 − 0.142i)19-s + (0.675 − 0.675i)21-s + (0.0879 + 0.328i)23-s + (0.832 − 0.554i)25-s + (−0.284 + 0.284i)27-s + (−0.448 − 0.259i)29-s + (1.31 + 1.31i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0308 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0308 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.0308 - 0.999i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.0308 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.548923 + 0.532265i\)
\(L(\frac12)\) \(\approx\) \(0.548923 + 0.532265i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.14 + 0.647i)T \)
13 \( 1 + (-0.727 - 3.53i)T \)
good3 \( 1 + (2.51 - 0.674i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (1.45 - 0.839i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.758 - 2.83i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.11 - 7.90i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.32 + 0.621i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.421 - 1.57i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (2.41 + 1.39i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-7.32 - 7.32i)T + 31iT^{2} \)
37 \( 1 + (3.60 + 2.08i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.77 + 1.28i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-2.93 - 0.786i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + 10.5iT - 47T^{2} \)
53 \( 1 + (6.66 + 6.66i)T + 53iT^{2} \)
59 \( 1 + (-1.16 + 4.35i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.16 + 3.75i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.47 - 4.28i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.63 + 6.10i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 - 7.13T + 73T^{2} \)
79 \( 1 - 9.73iT - 79T^{2} \)
83 \( 1 - 6.67iT - 83T^{2} \)
89 \( 1 + (0.578 - 0.154i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-6.03 - 10.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.34788181179377550302541097081, −11.17089972664366281757950176914, −10.36958760870488441494063407104, −9.612871187774388887268896010721, −8.621150888823959608617413089923, −6.63603795956958001454830766346, −6.27141551476724800219858878699, −5.16083559587079426590762559410, −4.14300812547770972605267770701, −1.83359451707130256991611079280, 0.72917733493378242421280956051, 2.86622303926468772478715386844, 4.81673895498100591641292435638, 5.92948595281983952144792864173, 6.39726053187964971553657108749, 7.48014301573963770107088073717, 9.085003355979627607552841494299, 10.10599683380562466162239021955, 10.91364839817077627331793460011, 11.59570129104773192211559802041

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