↑

Properties

Label 2-260-20.3-c1-0-21
Degree $2$
Conductor $260$
Sign $0.743 - 0.669i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.688 + 1.23i)2-s + (1.59 − 1.59i)3-s + (−1.05 + 1.70i)4-s + (1.94 + 1.10i)5-s + (3.06 + 0.871i)6-s + (−0.0942 − 0.0942i)7-s + (−2.82 − 0.127i)8-s − 2.08i·9-s + (−0.0230 + 3.16i)10-s − 2.08i·11-s + (1.03 + 4.38i)12-s + (−0.707 − 0.707i)13-s + (0.0515 − 0.181i)14-s + (4.85 − 1.34i)15-s + (−1.78 − 3.57i)16-s + (−3.21 + 3.21i)17-s + ⋯
L(s)  = 1  + (0.486 + 0.873i)2-s + (0.920 − 0.920i)3-s + (−0.525 + 0.850i)4-s + (0.869 + 0.493i)5-s + (1.25 + 0.355i)6-s + (−0.0356 − 0.0356i)7-s + (−0.998 − 0.0452i)8-s − 0.694i·9-s + (−0.00729 + 0.999i)10-s − 0.629i·11-s + (0.298 + 1.26i)12-s + (−0.196 − 0.196i)13-s + (0.0137 − 0.0484i)14-s + (1.25 − 0.346i)15-s + (−0.446 − 0.894i)16-s + (−0.779 + 0.779i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.743 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.743 - 0.669i$
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: \((2,\ 260,\ (\ :1/2),\ 0.743 - 0.669i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97052 + 0.756423i\)
\(L(\frac12)\) \(\approx\) \(1.97052 + 0.756423i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.688 - 1.23i)T \)
5 \( 1 + (-1.94 - 1.10i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-1.59 + 1.59i)T - 3iT^{2} \)
7 \( 1 + (0.0942 + 0.0942i)T + 7iT^{2} \)
11 \( 1 + 2.08iT - 11T^{2} \)
17 \( 1 + (3.21 - 3.21i)T - 17iT^{2} \)
19 \( 1 + 3.07T + 19T^{2} \)
23 \( 1 + (-4.44 + 4.44i)T - 23iT^{2} \)
29 \( 1 + 0.0603iT - 29T^{2} \)
31 \( 1 + 3.95iT - 31T^{2} \)
37 \( 1 + (3.14 - 3.14i)T - 37iT^{2} \)
41 \( 1 + 8.79T + 41T^{2} \)
43 \( 1 + (6.26 - 6.26i)T - 43iT^{2} \)
47 \( 1 + (5.57 + 5.57i)T + 47iT^{2} \)
53 \( 1 + (3.86 + 3.86i)T + 53iT^{2} \)
59 \( 1 - 5.43T + 59T^{2} \)
61 \( 1 - 12.0T + 61T^{2} \)
67 \( 1 + (5.51 + 5.51i)T + 67iT^{2} \)
71 \( 1 - 1.82iT - 71T^{2} \)
73 \( 1 + (-10.5 - 10.5i)T + 73iT^{2} \)
79 \( 1 - 7.53T + 79T^{2} \)
83 \( 1 + (10.7 - 10.7i)T - 83iT^{2} \)
89 \( 1 - 2.38iT - 89T^{2} \)
97 \( 1 + (-8.89 + 8.89i)T - 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.77216768877925457509918444501, −11.28320286542468469761462594028, −9.990955534553788948446064005520, −8.659669513675335232563027563195, −8.269435147763423857920590055397, −6.86566109043364128641880682110, −6.47835818365491548876041082808, −5.10670706024361801573363535262, −3.40407672811778423918898541993, −2.24146479883755632523229658369, 1.96787009547795274290694579592, 3.17227811921397030574981874208, 4.47787957324425684013058242023, 5.19823460895282850062290005451, 6.71689339229850300992036845889, 8.636890476509380316590835691218, 9.250587121456863925440529288737, 9.892094131198365877494706258641, 10.72894437998138751218696071788, 11.93877297109274922218045019931

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