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Properties

Label 2-490-245.103-c1-0-25
Degree $2$
Conductor $490$
Sign $-0.859 + 0.510i$
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  + (0.593 − 0.804i)2-s + (−0.130 + 0.0246i)3-s + (−0.294 − 0.955i)4-s + (1.13 − 1.92i)5-s + (−0.0575 + 0.119i)6-s + (−2.40 − 1.10i)7-s + (−0.943 − 0.330i)8-s + (−2.77 + 1.08i)9-s + (−0.871 − 2.05i)10-s + (1.22 − 3.11i)11-s + (0.0619 + 0.117i)12-s + (−4.17 − 0.469i)13-s + (−2.31 + 1.28i)14-s + (−0.101 + 0.278i)15-s + (−0.826 + 0.563i)16-s + (3.06 + 0.114i)17-s + ⋯
L(s)  = 1  + (0.419 − 0.568i)2-s + (−0.0751 + 0.0142i)3-s + (−0.147 − 0.477i)4-s + (0.509 − 0.860i)5-s + (−0.0234 + 0.0487i)6-s + (−0.909 − 0.416i)7-s + (−0.333 − 0.116i)8-s + (−0.925 + 0.363i)9-s + (−0.275 − 0.651i)10-s + (0.368 − 0.938i)11-s + (0.0178 + 0.0338i)12-s + (−1.15 − 0.130i)13-s + (−0.618 + 0.342i)14-s + (−0.0260 + 0.0719i)15-s + (−0.206 + 0.140i)16-s + (0.742 + 0.0277i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 + 0.510i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 + 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.859 + 0.510i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ -0.859 + 0.510i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.337029 - 1.22654i\)
\(L(\frac12)\) \(\approx\) \(0.337029 - 1.22654i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.593 + 0.804i)T \)
5 \( 1 + (-1.13 + 1.92i)T \)
7 \( 1 + (2.40 + 1.10i)T \)
good3 \( 1 + (0.130 - 0.0246i)T + (2.79 - 1.09i)T^{2} \)
11 \( 1 + (-1.22 + 3.11i)T + (-8.06 - 7.48i)T^{2} \)
13 \( 1 + (4.17 + 0.469i)T + (12.6 + 2.89i)T^{2} \)
17 \( 1 + (-3.06 - 0.114i)T + (16.9 + 1.27i)T^{2} \)
19 \( 1 + (-0.0882 + 0.152i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.00716 - 0.191i)T + (-22.9 + 1.71i)T^{2} \)
29 \( 1 + (-7.23 + 1.65i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (-1.68 + 0.971i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.39 - 0.736i)T + (20.8 - 30.5i)T^{2} \)
41 \( 1 + (0.318 + 0.661i)T + (-25.5 + 32.0i)T^{2} \)
43 \( 1 + (2.12 + 6.06i)T + (-33.6 + 26.8i)T^{2} \)
47 \( 1 + (7.83 + 5.78i)T + (13.8 + 44.9i)T^{2} \)
53 \( 1 + (-11.7 - 6.23i)T + (29.8 + 43.7i)T^{2} \)
59 \( 1 + (0.225 + 3.01i)T + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (-2.20 + 7.13i)T + (-50.4 - 34.3i)T^{2} \)
67 \( 1 + (-8.58 - 2.30i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.98 + 8.71i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (9.36 - 6.91i)T + (21.5 - 69.7i)T^{2} \)
79 \( 1 + (-6.30 - 3.64i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.484 + 4.29i)T + (-80.9 + 18.4i)T^{2} \)
89 \( 1 + (-6.30 - 16.0i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-7.95 - 7.95i)T + 97iT^{2} \)
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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

−10.47071892864563413756351155208, −9.882409834111782580980949919008, −8.971834813186949648624653858950, −8.089129437083753255472510678957, −6.61742429938633328445337467753, −5.68696398409460996698724019597, −4.92117484840515265715015711295, −3.57698447657832287074770911705, −2.48978322155962261333633705423, −0.65336808322676301852844405401, 2.51399593080019030130692876499, 3.35212816645752968658640351055, 4.88068161948560465592452996950, 5.91847788162904863573558221420, 6.64092306333664865137209573864, 7.35944252006181210766860463283, 8.648492751096263059016716463576, 9.685021465043827220864173304911, 10.14284659872401232515338500742, 11.61505106210124274261812638091

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