Properties

Label 2-490-245.239-c1-0-5
Degree $2$
Conductor $490$
Sign $0.965 - 0.260i$
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  + (−0.781 − 0.623i)2-s + (0.318 − 0.661i)3-s + (0.222 + 0.974i)4-s + (−2.09 − 0.794i)5-s + (−0.661 + 0.318i)6-s + (2.42 + 1.04i)7-s + (0.433 − 0.900i)8-s + (1.53 + 1.92i)9-s + (1.13 + 1.92i)10-s + (−3.47 + 4.35i)11-s + (0.715 + 0.163i)12-s + (2.13 + 1.70i)13-s + (−1.24 − 2.33i)14-s + (−1.19 + 1.12i)15-s + (−0.900 + 0.433i)16-s + (0.971 + 0.221i)17-s + ⋯
L(s)  = 1  + (−0.552 − 0.440i)2-s + (0.183 − 0.381i)3-s + (0.111 + 0.487i)4-s + (−0.934 − 0.355i)5-s + (−0.270 + 0.130i)6-s + (0.918 + 0.396i)7-s + (0.153 − 0.318i)8-s + (0.511 + 0.641i)9-s + (0.360 + 0.608i)10-s + (−1.04 + 1.31i)11-s + (0.206 + 0.0471i)12-s + (0.592 + 0.472i)13-s + (−0.332 − 0.623i)14-s + (−0.307 + 0.291i)15-s + (−0.225 + 0.108i)16-s + (0.235 + 0.0537i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.260i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.965 - 0.260i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ 0.965 - 0.260i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00995 + 0.134010i\)
\(L(\frac12)\) \(\approx\) \(1.00995 + 0.134010i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.781 + 0.623i)T \)
5 \( 1 + (2.09 + 0.794i)T \)
7 \( 1 + (-2.42 - 1.04i)T \)
good3 \( 1 + (-0.318 + 0.661i)T + (-1.87 - 2.34i)T^{2} \)
11 \( 1 + (3.47 - 4.35i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (-2.13 - 1.70i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-0.971 - 0.221i)T + (15.3 + 7.37i)T^{2} \)
19 \( 1 + 3.53T + 19T^{2} \)
23 \( 1 + (-4.50 + 1.02i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-0.0726 + 0.318i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 - 7.02T + 31T^{2} \)
37 \( 1 + (2.14 + 0.489i)T + (33.3 + 16.0i)T^{2} \)
41 \( 1 + (-2.19 - 1.05i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-5.13 - 10.6i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (6.06 + 4.83i)T + (10.4 + 45.8i)T^{2} \)
53 \( 1 + (3.19 - 0.729i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (0.581 - 0.279i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (0.124 - 0.544i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + 11.4iT - 67T^{2} \)
71 \( 1 + (-2.88 - 12.6i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-5.91 + 4.71i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 + (-5.15 + 4.11i)T + (18.4 - 80.9i)T^{2} \)
89 \( 1 + (4.20 + 5.27i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + 1.20iT - 97T^{2} \)
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

−10.99617684482828215360672086456, −10.26162537901017281451331172585, −9.072392436311522732286337919455, −8.104256164429676089653790313531, −7.79140286606932368408539722549, −6.77911437776419287788593651011, −4.94127792287267708390353705600, −4.35656748819957182545968190686, −2.61872488692869484145863960511, −1.48662002382580959512047605368, 0.814165875616495826284000808450, 3.02864511394565577165297605829, 4.12860537760902390505853701441, 5.26575605931801868554866860486, 6.48825912703159185466708597994, 7.51144592718712057507662363364, 8.267172001335563956769769168634, 8.801723870137962372312487589117, 10.22759232158853352052684259904, 10.80237446798376038770666914760

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