Properties

Label 2-490-35.13-c1-0-10
Degree $2$
Conductor $490$
Sign $-0.218 + 0.975i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.830 + 0.830i)3-s − 1.00i·4-s + (−2.05 + 0.880i)5-s − 1.17i·6-s + (0.707 + 0.707i)8-s + 1.62i·9-s + (0.830 − 2.07i)10-s + 0.743·11-s + (0.830 + 0.830i)12-s + (−2.05 + 2.05i)13-s + (0.975 − 2.43i)15-s − 1.00·16-s + (−4.63 − 4.63i)17-s + (−1.14 − 1.14i)18-s + 1.89·19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.479 + 0.479i)3-s − 0.500i·4-s + (−0.919 + 0.393i)5-s − 0.479i·6-s + (0.250 + 0.250i)8-s + 0.540i·9-s + (0.262 − 0.656i)10-s + 0.224·11-s + (0.239 + 0.239i)12-s + (−0.570 + 0.570i)13-s + (0.251 − 0.629i)15-s − 0.250·16-s + (−1.12 − 1.12i)17-s + (−0.270 − 0.270i)18-s + 0.434·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.218 + 0.975i$
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: \((2,\ 490,\ (\ :1/2),\ -0.218 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0209333 - 0.0261425i\)
\(L(\frac12)\) \(\approx\) \(0.0209333 - 0.0261425i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.05 - 0.880i)T \)
7 \( 1 \)
good3 \( 1 + (0.830 - 0.830i)T - 3iT^{2} \)
11 \( 1 - 0.743T + 11T^{2} \)
13 \( 1 + (2.05 - 2.05i)T - 13iT^{2} \)
17 \( 1 + (4.63 + 4.63i)T + 17iT^{2} \)
19 \( 1 - 1.89T + 19T^{2} \)
23 \( 1 + (3.74 + 3.74i)T + 23iT^{2} \)
29 \( 1 + 9.69iT - 29T^{2} \)
31 \( 1 - 3.42iT - 31T^{2} \)
37 \( 1 + (1.88 - 1.88i)T - 37iT^{2} \)
41 \( 1 + 0.817iT - 41T^{2} \)
43 \( 1 + (-1.59 - 1.59i)T + 43iT^{2} \)
47 \( 1 + (3.33 + 3.33i)T + 47iT^{2} \)
53 \( 1 + (-3.52 - 3.52i)T + 53iT^{2} \)
59 \( 1 + 2.54T + 59T^{2} \)
61 \( 1 + 6.07iT - 61T^{2} \)
67 \( 1 + (9.68 - 9.68i)T - 67iT^{2} \)
71 \( 1 + 16.0T + 71T^{2} \)
73 \( 1 + (-6.25 + 6.25i)T - 73iT^{2} \)
79 \( 1 + 6.58iT - 79T^{2} \)
83 \( 1 + (9.23 - 9.23i)T - 83iT^{2} \)
89 \( 1 - 6.03T + 89T^{2} \)
97 \( 1 + (3.16 + 3.16i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.63597066169897888567156964861, −9.880781347481200701308023200545, −8.891999643683342438238614634894, −7.892764686462358513723595871928, −7.13518050512361212645895542041, −6.22224167783721176580755769397, −4.88600618202249510950569970891, −4.22688579921651283834851925644, −2.46249042916843609400826222463, −0.02562902663432083958228799685, 1.50275644696088147352417613253, 3.29845559017632398583215677274, 4.30172540758865505681661260987, 5.65268436816456319144506323963, 6.86502167435205456924176400365, 7.63132488238807806647586184249, 8.593386326740474789108621143695, 9.353573595659171811595322024485, 10.50687868122622387034253750739, 11.31025357837085368796929675368

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