Properties

Label 2-490-35.4-c1-0-18
Degree $2$
Conductor $490$
Sign $-0.330 + 0.943i$
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 + (−0.133 − 2.23i)5-s − 0.999i·8-s + (−1.5 − 2.59i)9-s + (−1.23 − 1.86i)10-s + (−1.5 + 2.59i)11-s − 5i·13-s + (−0.5 − 0.866i)16-s + (−1.73 − i)17-s + (−2.59 − 1.5i)18-s + (2.5 + 4.33i)19-s + (−1.99 − 0.999i)20-s + 3i·22-s + (6.06 − 3.5i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.0599 − 0.998i)5-s − 0.353i·8-s + (−0.5 − 0.866i)9-s + (−0.389 − 0.590i)10-s + (−0.452 + 0.783i)11-s − 1.38i·13-s + (−0.125 − 0.216i)16-s + (−0.420 − 0.242i)17-s + (−0.612 − 0.353i)18-s + (0.573 + 0.993i)19-s + (−0.447 − 0.223i)20-s + 0.639i·22-s + (1.26 − 0.729i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.330 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.330 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.330 + 0.943i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ -0.330 + 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01718 - 1.43404i\)
\(L(\frac12)\) \(\approx\) \(1.01718 - 1.43404i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.133 + 2.23i)T \)
7 \( 1 \)
good3 \( 1 + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5iT - 13T^{2} \)
17 \( 1 + (1.73 + i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.06 + 3.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.866 + 0.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + (-6.06 + 3.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.79 - 4.5i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.73 + i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-13.8 - 8i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7 - 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + (1 + 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12iT - 97T^{2} \)
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

−10.71065327786033424938821959137, −9.912131235623781547540789915505, −8.956854497953787898374388494823, −8.074746369172733590685425481663, −6.91120559008235542510847857684, −5.64696656141744043016714341504, −5.03938790806980925456731354160, −3.84995930337431560083307506826, −2.66847739324534965920257532387, −0.896651004109062042349604060252, 2.35521198192347087251404604157, 3.30458746709726134917505246674, 4.62703886472197383041144228885, 5.62989690338705083596775032238, 6.67131875261339353571096160873, 7.35075183883875532309252744906, 8.390350618638631889355947715403, 9.369248047316683462904369112230, 10.74446310894925972923768427073, 11.20431354138504903109650998309

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