↑

Properties

Label 2-490-35.13-c1-0-2
Degree $2$
Conductor $490$
Sign $-0.753 - 0.657i$
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  + (0.707 − 0.707i)2-s + (−1.42 + 1.42i)3-s − 1.00i·4-s + (−0.204 + 2.22i)5-s + 2.02i·6-s + (−0.707 − 0.707i)8-s − 1.08i·9-s + (1.42 + 1.71i)10-s − 4.03·11-s + (1.42 + 1.42i)12-s + (−0.204 + 0.204i)13-s + (−2.89 − 3.47i)15-s − 1.00·16-s + (−1.44 − 1.44i)17-s + (−0.769 − 0.769i)18-s − 6.20·19-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.825 + 0.825i)3-s − 0.500i·4-s + (−0.0916 + 0.995i)5-s + 0.825i·6-s + (−0.250 − 0.250i)8-s − 0.362i·9-s + (0.452 + 0.543i)10-s − 1.21·11-s + (0.412 + 0.412i)12-s + (−0.0568 + 0.0568i)13-s + (−0.746 − 0.897i)15-s − 0.250·16-s + (−0.349 − 0.349i)17-s + (−0.181 − 0.181i)18-s − 1.42·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.753 - 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.753 - 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.753 - 0.657i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ -0.753 - 0.657i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.232663 + 0.620946i\)
\(L(\frac12)\) \(\approx\) \(0.232663 + 0.620946i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (0.204 - 2.22i)T \)
7 \( 1 \)
good3 \( 1 + (1.42 - 1.42i)T - 3iT^{2} \)
11 \( 1 + 4.03T + 11T^{2} \)
13 \( 1 + (0.204 - 0.204i)T - 13iT^{2} \)
17 \( 1 + (1.44 + 1.44i)T + 17iT^{2} \)
19 \( 1 + 6.20T + 19T^{2} \)
23 \( 1 + (-3.20 - 3.20i)T + 23iT^{2} \)
29 \( 1 - 7.15iT - 29T^{2} \)
31 \( 1 - 7.31iT - 31T^{2} \)
37 \( 1 + (-3.27 + 3.27i)T - 37iT^{2} \)
41 \( 1 - 2.58iT - 41T^{2} \)
43 \( 1 + (4.97 + 4.97i)T + 43iT^{2} \)
47 \( 1 + (-0.222 - 0.222i)T + 47iT^{2} \)
53 \( 1 + (-5.85 - 5.85i)T + 53iT^{2} \)
59 \( 1 - 0.855T + 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 + (2.23 - 2.23i)T - 67iT^{2} \)
71 \( 1 - 7.12T + 71T^{2} \)
73 \( 1 + (-8.15 + 8.15i)T - 73iT^{2} \)
79 \( 1 + 5.07iT - 79T^{2} \)
83 \( 1 + (-3.85 + 3.85i)T - 83iT^{2} \)
89 \( 1 + 3.07T + 89T^{2} \)
97 \( 1 + (-6.63 - 6.63i)T + 97iT^{2} \)
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

−10.98762178023619671901594525071, −10.67261835461997985548234319304, −10.08336021366125072623525654124, −8.856391176011624410600257395168, −7.47287994794345041672069344141, −6.47398425554004116587100103110, −5.42532695693603347375900974223, −4.69936925178922480295627237019, −3.50881970112170355533823259403, −2.36459909135410225334741138048, 0.35151834417837811716797330220, 2.25067714102503716781817096060, 4.14082843272193025551189949419, 5.08510035224071110199172436760, 5.95526633683735641028154961821, 6.70718372335894414555764939396, 7.88360921395023841513785944305, 8.414722974283646052827053926425, 9.686653055306106615003052254508, 10.93468956852386803875519229140

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