↑

Properties

Label 2-490-35.9-c1-0-19
Degree $2$
Conductor $490$
Sign $0.830 + 0.556i$
Analytic cond. $3.91266$
Root an. cond. $1.97804$
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  + (0.866 + 0.5i)2-s + (2.59 − 1.5i)3-s + (0.499 + 0.866i)4-s + (−1.23 − 1.86i)5-s + 3·6-s + 0.999i·8-s + (3 − 5.19i)9-s + (−0.133 − 2.23i)10-s + (2.59 + 1.50i)12-s + 2i·13-s + (−6 − 3i)15-s + (−0.5 + 0.866i)16-s + (−1.73 + i)17-s + (5.19 − 3i)18-s + (1 − 1.73i)19-s + (1 − 1.99i)20-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (1.49 − 0.866i)3-s + (0.249 + 0.433i)4-s + (−0.550 − 0.834i)5-s + 1.22·6-s + 0.353i·8-s + (1 − 1.73i)9-s + (−0.0423 − 0.705i)10-s + (0.749 + 0.433i)12-s + 0.554i·13-s + (−1.54 − 0.774i)15-s + (−0.125 + 0.216i)16-s + (−0.420 + 0.242i)17-s + (1.22 − 0.707i)18-s + (0.229 − 0.397i)19-s + (0.223 − 0.447i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.830 + 0.556i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ 0.830 + 0.556i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.75997 - 0.839264i\)
\(L(\frac12)\) \(\approx\) \(2.75997 - 0.839264i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (1.23 + 1.86i)T \)
7 \( 1 \)
good3 \( 1 + (-2.59 + 1.5i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + (1.73 - i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + (-5 - 8.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.92 + 4i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 - 5iT - 43T^{2} \)
47 \( 1 + (6.92 + 4i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.19 - 3i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1 + 1.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.5 - 7.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.06 + 3.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (-8.66 + 5i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5 - 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9iT - 83T^{2} \)
89 \( 1 + (-3.5 + 6.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 97T^{2} \)
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

−11.15224408610367439682641640415, −9.556967743266811028641887810856, −8.717407519514583404627009262517, −8.212002623633359773729200142772, −7.28215191276137394092523571041, −6.56490203397105135433768481944, −5.00118432280535199166807086174, −3.95106437028853899356553115478, −2.93846823101421540246862549715, −1.56559820086775594019784258294, 2.33423728391999526839941474582, 3.19718156470117231343074168530, 3.94005557007521707030496466276, 4.93192999322142060820396217217, 6.44921517520472811182374058573, 7.64559655388369570846117701483, 8.309851491065258016233971174370, 9.465954306097583507715565831752, 10.16127692358700866687085143737, 10.89892148996807018289167541845

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