↑

Properties

Label 2-490-245.239-c1-0-24
Degree $2$
Conductor $490$
Sign $-0.854 + 0.518i$
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 + (1.36 − 2.82i)3-s + (0.222 + 0.974i)4-s + (1.18 − 1.89i)5-s + (−2.82 + 1.36i)6-s + (1.52 + 2.16i)7-s + (0.433 − 0.900i)8-s + (−4.27 − 5.35i)9-s + (−2.10 + 0.747i)10-s + (−0.773 + 0.969i)11-s + (3.05 + 0.698i)12-s + (−4.30 − 3.43i)13-s + (0.153 − 2.64i)14-s + (−3.75 − 5.92i)15-s + (−0.900 + 0.433i)16-s + (1.89 + 0.431i)17-s + ⋯
L(s)  = 1  + (−0.552 − 0.440i)2-s + (0.786 − 1.63i)3-s + (0.111 + 0.487i)4-s + (0.528 − 0.848i)5-s + (−1.15 + 0.555i)6-s + (0.577 + 0.816i)7-s + (0.153 − 0.318i)8-s + (−1.42 − 1.78i)9-s + (−0.666 + 0.236i)10-s + (−0.233 + 0.292i)11-s + (0.883 + 0.201i)12-s + (−1.19 − 0.951i)13-s + (0.0409 − 0.705i)14-s + (−0.970 − 1.53i)15-s + (−0.225 + 0.108i)16-s + (0.458 + 0.104i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 + 0.518i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.854 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.854 + 0.518i$
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: \((2,\ 490,\ (\ :1/2),\ -0.854 + 0.518i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.396594 - 1.41784i\)
\(L(\frac12)\) \(\approx\) \(0.396594 - 1.41784i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.18 + 1.89i)T \)
7 \( 1 + (-1.52 - 2.16i)T \)
good3 \( 1 + (-1.36 + 2.82i)T + (-1.87 - 2.34i)T^{2} \)
11 \( 1 + (0.773 - 0.969i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (4.30 + 3.43i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-1.89 - 0.431i)T + (15.3 + 7.37i)T^{2} \)
19 \( 1 - 2.48T + 19T^{2} \)
23 \( 1 + (-3.40 + 0.778i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (0.423 - 1.85i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + 0.154T + 31T^{2} \)
37 \( 1 + (5.27 + 1.20i)T + (33.3 + 16.0i)T^{2} \)
41 \( 1 + (-6.23 - 3.00i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-0.374 - 0.778i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (-7.75 - 6.18i)T + (10.4 + 45.8i)T^{2} \)
53 \( 1 + (-10.8 + 2.47i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (13.2 - 6.37i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (2.42 - 10.6i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + 10.7iT - 67T^{2} \)
71 \( 1 + (2.64 + 11.6i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (5.96 - 4.75i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 + (-7.66 + 6.11i)T + (18.4 - 80.9i)T^{2} \)
89 \( 1 + (-7.92 - 9.93i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 1.61iT - 97T^{2} \)
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

−10.51227031377520024069226179000, −9.282518850913291204080159533986, −8.857666043986030063110799662906, −7.79473077623178621499376461301, −7.49188977069442568950833364436, −6.03113678032818171139527022801, −5.02749506384205548728944113136, −2.92597073407988327048994612181, −2.13966009304751794766919779322, −1.02346881040083641346675177052, 2.29944892456207598058126409441, 3.51733393366803663066851613271, 4.67103872992747020078267523710, 5.52695313007064167260284620758, 7.06818726624858885182925318537, 7.74531860545725322183987199178, 8.936118854248995601486441411146, 9.573006951097317470535690616908, 10.28897013639647299168452051419, 10.79100515864675853506555578094

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