↑

Properties

Label 2-490-5.4-c3-0-53
Degree $2$
Conductor $490$
Sign $-0.766 + 0.642i$
Analytic cond. $28.9109$
Root an. cond. $5.37688$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s − 6.02i·3-s − 4·4-s + (7.18 + 8.56i)5-s + 12.0·6-s − 8i·8-s − 9.30·9-s + (−17.1 + 14.3i)10-s − 54.9·11-s + 24.0i·12-s + 10.6i·13-s + (51.6 − 43.3i)15-s + 16·16-s − 12.1i·17-s − 18.6i·18-s − 27.4·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.15i·3-s − 0.5·4-s + (0.642 + 0.766i)5-s + 0.819·6-s − 0.353i·8-s − 0.344·9-s + (−0.541 + 0.454i)10-s − 1.50·11-s + 0.579i·12-s + 0.227i·13-s + (0.888 − 0.745i)15-s + 0.250·16-s − 0.173i·17-s − 0.243i·18-s − 0.331·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(28.9109\)
Root analytic conductor: \(5.37688\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :3/2),\ -0.766 + 0.642i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4682808907\)
\(L(\frac12)\) \(\approx\) \(0.4682808907\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
5 \( 1 + (-7.18 - 8.56i)T \)
7 \( 1 \)
good3 \( 1 + 6.02iT - 27T^{2} \)
11 \( 1 + 54.9T + 1.33e3T^{2} \)
13 \( 1 - 10.6iT - 2.19e3T^{2} \)
17 \( 1 + 12.1iT - 4.91e3T^{2} \)
19 \( 1 + 27.4T + 6.85e3T^{2} \)
23 \( 1 + 181. iT - 1.21e4T^{2} \)
29 \( 1 - 124.T + 2.43e4T^{2} \)
31 \( 1 + 334.T + 2.97e4T^{2} \)
37 \( 1 + 88.1iT - 5.06e4T^{2} \)
41 \( 1 + 134.T + 6.89e4T^{2} \)
43 \( 1 - 190. iT - 7.95e4T^{2} \)
47 \( 1 - 177. iT - 1.03e5T^{2} \)
53 \( 1 + 211. iT - 1.48e5T^{2} \)
59 \( 1 + 342.T + 2.05e5T^{2} \)
61 \( 1 + 659.T + 2.26e5T^{2} \)
67 \( 1 + 572. iT - 3.00e5T^{2} \)
71 \( 1 + 743.T + 3.57e5T^{2} \)
73 \( 1 + 1.16e3iT - 3.89e5T^{2} \)
79 \( 1 + 1.00e3T + 4.93e5T^{2} \)
83 \( 1 - 348. iT - 5.71e5T^{2} \)
89 \( 1 - 1.06e3T + 7.04e5T^{2} \)
97 \( 1 - 578. iT - 9.12e5T^{2} \)
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

−10.29320121975352828946594114680, −9.130414788518967633560680403488, −8.033127889666811268940762732624, −7.37211832241117560010202260083, −6.58249192526011552485783635421, −5.86424901099888736904673393444, −4.71238482827402738631656191797, −2.92486697780801982492960270349, −1.88429259392396554660177008178, −0.13696211299731062784674665061, 1.64336918490181066341025181156, 3.01422304208664617204679343898, 4.15947028355258831376357038324, 5.14859943884203072189947117727, 5.64021679985127167916221181563, 7.49797466974000797609472985184, 8.618273433858758503029411329564, 9.330318459187377472717493480375, 10.19156221268739780627415331525, 10.53633664505248842200355726829

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