↑

Properties

Label 2-490-35.9-c1-0-5
Degree $2$
Conductor $490$
Sign $-0.246 - 0.969i$
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 + (0.499 + 0.866i)4-s + (−2.19 − 0.448i)5-s + 0.999i·8-s + (−1.5 + 2.59i)9-s + (−1.67 − 1.48i)10-s + (2 + 3.46i)11-s + 4.24i·13-s + (−0.5 + 0.866i)16-s + (3.67 − 2.12i)17-s + (−2.59 + 1.5i)18-s + (−2.82 + 4.89i)19-s + (−0.707 − 2.12i)20-s + 3.99i·22-s + (4.59 + 1.96i)25-s + (−2.12 + 3.67i)26-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.979 − 0.200i)5-s + 0.353i·8-s + (−0.5 + 0.866i)9-s + (−0.529 − 0.469i)10-s + (0.603 + 1.04i)11-s + 1.17i·13-s + (−0.125 + 0.216i)16-s + (0.891 − 0.514i)17-s + (−0.612 + 0.353i)18-s + (−0.648 + 1.12i)19-s + (−0.158 − 0.474i)20-s + 0.852i·22-s + (0.919 + 0.392i)25-s + (−0.416 + 0.720i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.246 - 0.969i$
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: \((2,\ 490,\ (\ :1/2),\ -0.246 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.923618 + 1.18801i\)
\(L(\frac12)\) \(\approx\) \(0.923618 + 1.18801i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.19 + 0.448i)T \)
7 \( 1 \)
good3 \( 1 + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.24iT - 13T^{2} \)
17 \( 1 + (-3.67 + 2.12i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.82 - 4.89i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (2.82 + 4.89i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.19 + 3i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.41T + 41T^{2} \)
43 \( 1 + 12iT - 43T^{2} \)
47 \( 1 + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (10.3 - 6i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.65 - 9.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.53 + 6.12i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.3 + 6i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (3.67 - 2.12i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 16.9iT - 83T^{2} \)
89 \( 1 + (-2.12 + 3.67i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.24iT - 97T^{2} \)
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

−11.50671529557614710800130365784, −10.50014376298182370696465388736, −9.300992755405197279220301286038, −8.297909455311541037052044432675, −7.51141168466346644861524794384, −6.72782025922558308844980564944, −5.43893793763337942888231757605, −4.46843895622302966768504186026, −3.68806037444316483404578936004, −2.06242027240983658224898531033, 0.77000627600658959443498098976, 3.09704937886795151950199254526, 3.52800212076404499604615505075, 4.86170995054942863367183419711, 6.02303749702719412340202545865, 6.81089707745841915736830574382, 8.119936144651313031012720037258, 8.775091753594705387054698742605, 10.04477000678638452993412988392, 11.03853750636298828488943121088

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