↑

Properties

Label 2-260-260.3-c1-0-27
Degree $2$
Conductor $260$
Sign $0.445 + 0.895i$
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  + (−0.524 − 1.31i)2-s + (2.57 + 0.689i)3-s + (−1.45 + 1.37i)4-s + (0.222 − 2.22i)5-s + (−0.443 − 3.73i)6-s + (1.04 − 0.280i)7-s + (2.56 + 1.18i)8-s + (3.54 + 2.04i)9-s + (−3.03 + 0.874i)10-s + (0.213 − 0.123i)11-s + (−4.67 + 2.54i)12-s + (−3.26 − 1.52i)13-s + (−0.915 − 1.22i)14-s + (2.10 − 5.56i)15-s + (0.206 − 3.99i)16-s + (1.56 + 5.82i)17-s + ⋯
L(s)  = 1  + (−0.370 − 0.928i)2-s + (1.48 + 0.397i)3-s + (−0.725 + 0.688i)4-s + (0.0995 − 0.995i)5-s + (−0.180 − 1.52i)6-s + (0.395 − 0.105i)7-s + (0.908 + 0.418i)8-s + (1.18 + 0.681i)9-s + (−0.961 + 0.276i)10-s + (0.0643 − 0.0371i)11-s + (−1.35 + 0.733i)12-s + (−0.906 − 0.422i)13-s + (−0.244 − 0.327i)14-s + (0.543 − 1.43i)15-s + (0.0516 − 0.998i)16-s + (0.378 + 1.41i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.445 + 0.895i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.445 + 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35605 - 0.840026i\)
\(L(\frac12)\) \(\approx\) \(1.35605 - 0.840026i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.524 + 1.31i)T \)
5 \( 1 + (-0.222 + 2.22i)T \)
13 \( 1 + (3.26 + 1.52i)T \)
good3 \( 1 + (-2.57 - 0.689i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-1.04 + 0.280i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.213 + 0.123i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.56 - 5.82i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.97 + 3.42i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.77 - 0.742i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (3.76 - 2.17i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.881iT - 31T^{2} \)
37 \( 1 + (4.95 + 1.32i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-5.89 - 10.2i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.517 + 1.93i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (6.04 + 6.04i)T + 47iT^{2} \)
53 \( 1 + (0.436 + 0.436i)T + 53iT^{2} \)
59 \( 1 + (5.04 - 8.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.71 - 4.70i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.36 - 12.5i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-8.10 - 4.68i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.41 + 3.41i)T + 73iT^{2} \)
79 \( 1 - 15.3T + 79T^{2} \)
83 \( 1 + (-10.8 + 10.8i)T - 83iT^{2} \)
89 \( 1 + (-1.31 + 0.762i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.34 + 12.4i)T + (-84.0 + 48.5i)T^{2} \)
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

−11.88379548043106006696248609851, −10.63129765722575529585439812568, −9.693068258146650548397732970278, −9.047970656506363949101460353848, −8.266992869924163208439914636980, −7.57318270214342619555941830030, −5.14466984540556197446458680091, −4.13255390299926493227827078554, −2.99471295932384229966964910600, −1.62825674677798347811662276268, 2.08660739412238997345147870416, 3.46757591587227668818044542930, 5.08309442939475310658101030196, 6.60909987441765280520191395572, 7.50774548195082812568048145726, 7.927249054067070208129668300731, 9.311258414580277252853711200581, 9.652334570523891138642922183425, 10.99094791180273421853300420139, 12.35735328260408480888052218776

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