Properties

Label 2-260-65.58-c1-0-3
Degree $2$
Conductor $260$
Sign $0.836 - 0.547i$
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  + (1.86 − 0.5i)3-s + (−1 + 2i)5-s + (2.23 + 3.86i)7-s + (0.633 − 0.366i)9-s + (−2.86 + 0.767i)11-s + (3 − 2i)13-s + (−0.866 + 4.23i)15-s + (0.866 − 3.23i)17-s + (1.13 − 4.23i)19-s + (6.09 + 6.09i)21-s + (−1.86 − 6.96i)23-s + (−3 − 4i)25-s + (−3.09 + 3.09i)27-s + (1.5 + 0.866i)29-s + (5.19 − 5.19i)31-s + ⋯
L(s)  = 1  + (1.07 − 0.288i)3-s + (−0.447 + 0.894i)5-s + (0.843 + 1.46i)7-s + (0.211 − 0.122i)9-s + (−0.864 + 0.231i)11-s + (0.832 − 0.554i)13-s + (−0.223 + 1.09i)15-s + (0.210 − 0.783i)17-s + (0.260 − 0.970i)19-s + (1.33 + 1.33i)21-s + (−0.389 − 1.45i)23-s + (−0.600 − 0.800i)25-s + (−0.596 + 0.596i)27-s + (0.278 + 0.160i)29-s + (0.933 − 0.933i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 - 0.547i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.836 - 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.836 - 0.547i$
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: \((2,\ 260,\ (\ :1/2),\ 0.836 - 0.547i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60710 + 0.479024i\)
\(L(\frac12)\) \(\approx\) \(1.60710 + 0.479024i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 2i)T \)
13 \( 1 + (-3 + 2i)T \)
good3 \( 1 + (-1.86 + 0.5i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (-2.23 - 3.86i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.86 - 0.767i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.866 + 3.23i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.13 + 4.23i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.86 + 6.96i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.5 - 0.866i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.19 + 5.19i)T - 31iT^{2} \)
37 \( 1 + (4.23 - 7.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.669 - 2.5i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-3.59 - 0.964i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 2.92T + 47T^{2} \)
53 \( 1 + (4.46 + 4.46i)T + 53iT^{2} \)
59 \( 1 + (-6.33 - 1.69i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-4.5 - 7.79i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (13.6 + 7.86i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.59 + 1.23i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + 1.07iT - 73T^{2} \)
79 \( 1 + 7.46iT - 79T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 + (0.794 + 2.96i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-3.69 + 2.13i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.96565310257565593193952640416, −11.25125199180344965795413134140, −10.18276678814458206686959655026, −8.822929390202160325820307593568, −8.253779384272495290365209100204, −7.47796226374428617936103785903, −6.08927578737864437354592114204, −4.80972659012944149074876397327, −2.96304883255103036573203004353, −2.43032236749597662862668088796, 1.47360264029622721508204285114, 3.59446630296290096976854578788, 4.21770592090254312313077881580, 5.60760672594298418688906956480, 7.45762509760061540198987733183, 8.097190214778718078640450167220, 8.757723796705884927624622649650, 9.962423034625890182360022266476, 10.85640098871989739182645696355, 11.86701497832908435500959430038

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