↑

Properties

Label 2-490-245.223-c1-0-0
Degree $2$
Conductor $490$
Sign $0.385 - 0.922i$
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  + (0.330 − 0.943i)2-s + (−1.35 − 2.15i)3-s + (−0.781 − 0.623i)4-s + (−2.16 + 0.542i)5-s + (−2.47 + 0.565i)6-s + (−2.63 − 0.178i)7-s + (−0.846 + 0.532i)8-s + (−1.50 + 3.12i)9-s + (−0.204 + 2.22i)10-s + (3.97 − 1.91i)11-s + (−0.284 + 2.52i)12-s + (−0.698 + 1.99i)13-s + (−1.04 + 2.43i)14-s + (4.10 + 3.93i)15-s + (0.222 + 0.974i)16-s + (−0.167 + 1.48i)17-s + ⋯
L(s)  = 1  + (0.233 − 0.667i)2-s + (−0.781 − 1.24i)3-s + (−0.390 − 0.311i)4-s + (−0.970 + 0.242i)5-s + (−1.01 + 0.231i)6-s + (−0.997 − 0.0674i)7-s + (−0.299 + 0.188i)8-s + (−0.501 + 1.04i)9-s + (−0.0646 + 0.704i)10-s + (1.19 − 0.577i)11-s + (−0.0821 + 0.729i)12-s + (−0.193 + 0.553i)13-s + (−0.278 + 0.650i)14-s + (1.05 + 1.01i)15-s + (0.0556 + 0.243i)16-s + (−0.0406 + 0.360i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.385 - 0.922i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ 0.385 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00357518 + 0.00238183i\)
\(L(\frac12)\) \(\approx\) \(0.00357518 + 0.00238183i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.330 + 0.943i)T \)
5 \( 1 + (2.16 - 0.542i)T \)
7 \( 1 + (2.63 + 0.178i)T \)
good3 \( 1 + (1.35 + 2.15i)T + (-1.30 + 2.70i)T^{2} \)
11 \( 1 + (-3.97 + 1.91i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (0.698 - 1.99i)T + (-10.1 - 8.10i)T^{2} \)
17 \( 1 + (0.167 - 1.48i)T + (-16.5 - 3.78i)T^{2} \)
19 \( 1 + 2.78T + 19T^{2} \)
23 \( 1 + (-2.84 + 0.320i)T + (22.4 - 5.11i)T^{2} \)
29 \( 1 + (6.67 - 5.31i)T + (6.45 - 28.2i)T^{2} \)
31 \( 1 - 3.42iT - 31T^{2} \)
37 \( 1 + (5.95 + 0.670i)T + (36.0 + 8.23i)T^{2} \)
41 \( 1 + (-2.70 - 0.617i)T + (36.9 + 17.7i)T^{2} \)
43 \( 1 + (-3.00 + 4.78i)T + (-18.6 - 38.7i)T^{2} \)
47 \( 1 + (10.7 + 3.76i)T + (36.7 + 29.3i)T^{2} \)
53 \( 1 + (12.9 - 1.46i)T + (51.6 - 11.7i)T^{2} \)
59 \( 1 + (-2.99 - 13.1i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (-0.745 + 0.594i)T + (13.5 - 59.4i)T^{2} \)
67 \( 1 + (0.535 - 0.535i)T - 67iT^{2} \)
71 \( 1 + (0.609 - 0.764i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (6.02 - 2.10i)T + (57.0 - 45.5i)T^{2} \)
79 \( 1 - 5.02iT - 79T^{2} \)
83 \( 1 + (-3.85 + 1.34i)T + (64.8 - 51.7i)T^{2} \)
89 \( 1 + (14.5 + 7.01i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (-4.51 - 4.51i)T + 97iT^{2} \)
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

−11.29273318495076737251790975545, −10.72006315497817039644359251277, −9.345274946282821428672930586051, −8.478449770105164767544929074171, −7.05542366222400027221051840789, −6.71631578415598937473593054581, −5.67878423228474853087387017321, −4.15266782071031343899216617358, −3.19783763169776609917446869395, −1.49038270739123656826540216869, 0.00275944712829301808399007301, 3.40730775846569415134969595994, 4.17364911526010752137269057417, 4.96734501930859132280723420712, 6.07482624993724766278842599002, 6.92591804310725323764466177550, 8.027376207463584132854723037053, 9.349628619553778865999198142143, 9.603347468600726675592223901168, 10.87613686757484148954667824835

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