Properties

Label 2-260-20.3-c1-0-26
Degree $2$
Conductor $260$
Sign $-0.490 + 0.871i$
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  + (−1.08 − 0.904i)2-s + (1.78 − 1.78i)3-s + (0.364 + 1.96i)4-s + (−1.34 − 1.78i)5-s + (−3.56 + 0.327i)6-s + (2.34 + 2.34i)7-s + (1.38 − 2.46i)8-s − 3.39i·9-s + (−0.160 + 3.15i)10-s − 5.90i·11-s + (4.16 + 2.86i)12-s + (−0.707 − 0.707i)13-s + (−0.429 − 4.66i)14-s + (−5.59 − 0.803i)15-s + (−3.73 + 1.43i)16-s + (−0.975 + 0.975i)17-s + ⋯
L(s)  = 1  + (−0.768 − 0.639i)2-s + (1.03 − 1.03i)3-s + (0.182 + 0.983i)4-s + (−0.599 − 0.800i)5-s + (−1.45 + 0.133i)6-s + (0.886 + 0.886i)7-s + (0.488 − 0.872i)8-s − 1.13i·9-s + (−0.0508 + 0.998i)10-s − 1.78i·11-s + (1.20 + 0.826i)12-s + (−0.196 − 0.196i)13-s + (−0.114 − 1.24i)14-s + (−1.44 − 0.207i)15-s + (−0.933 + 0.358i)16-s + (−0.236 + 0.236i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.490 + 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.490 + 0.871i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.490 + 0.871i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.583541 - 0.998212i\)
\(L(\frac12)\) \(\approx\) \(0.583541 - 0.998212i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.08 + 0.904i)T \)
5 \( 1 + (1.34 + 1.78i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-1.78 + 1.78i)T - 3iT^{2} \)
7 \( 1 + (-2.34 - 2.34i)T + 7iT^{2} \)
11 \( 1 + 5.90iT - 11T^{2} \)
17 \( 1 + (0.975 - 0.975i)T - 17iT^{2} \)
19 \( 1 - 2.12T + 19T^{2} \)
23 \( 1 + (3.10 - 3.10i)T - 23iT^{2} \)
29 \( 1 - 5.56iT - 29T^{2} \)
31 \( 1 + 7.23iT - 31T^{2} \)
37 \( 1 + (2.19 - 2.19i)T - 37iT^{2} \)
41 \( 1 + 0.538T + 41T^{2} \)
43 \( 1 + (-5.78 + 5.78i)T - 43iT^{2} \)
47 \( 1 + (-3.98 - 3.98i)T + 47iT^{2} \)
53 \( 1 + (-7.09 - 7.09i)T + 53iT^{2} \)
59 \( 1 + 3.05T + 59T^{2} \)
61 \( 1 - 8.82T + 61T^{2} \)
67 \( 1 + (-2.58 - 2.58i)T + 67iT^{2} \)
71 \( 1 - 8.53iT - 71T^{2} \)
73 \( 1 + (-2.76 - 2.76i)T + 73iT^{2} \)
79 \( 1 - 7.02T + 79T^{2} \)
83 \( 1 + (6.93 - 6.93i)T - 83iT^{2} \)
89 \( 1 + 9.42iT - 89T^{2} \)
97 \( 1 + (11.8 - 11.8i)T - 97iT^{2} \)
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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

−11.74193060660748161341162921363, −10.98586290491157892816994908671, −9.289775102032603939794655247701, −8.525353340308057119863554463915, −8.209473952348495368615963479083, −7.33172713361360691673459500340, −5.59034800330725087886817383011, −3.74414206922008587412146424032, −2.50964984841905087683347890033, −1.16871078094218708796088687432, 2.27386559279023884245628996018, 4.07964965733386882527448880138, 4.81101443620526088706915947704, 6.85161609114132777780156346304, 7.56898762206394326319848459430, 8.326956458491769481173218614270, 9.522908966693693689794142531579, 10.17856332829570188802220765379, 10.83697459718342418621346054619, 12.00363236307481837753317111219

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