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Properties

Label 2-490-35.17-c1-0-19
Degree $2$
Conductor $490$
Sign $-0.205 + 0.978i$
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  + (0.258 − 0.965i)2-s + (3.10 − 0.830i)3-s + (−0.866 − 0.499i)4-s + (−0.475 − 2.18i)5-s − 3.20i·6-s + (−0.707 + 0.707i)8-s + (6.32 − 3.65i)9-s + (−2.23 − 0.106i)10-s + (−1.22 + 2.12i)11-s + (−3.10 − 0.830i)12-s + (0.884 + 0.884i)13-s + (−3.28 − 6.37i)15-s + (0.500 + 0.866i)16-s + (1.25 + 4.69i)17-s + (−1.89 − 7.05i)18-s + (0.522 + 0.904i)19-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (1.79 − 0.479i)3-s + (−0.433 − 0.249i)4-s + (−0.212 − 0.977i)5-s − 1.31i·6-s + (−0.249 + 0.249i)8-s + (2.10 − 1.21i)9-s + (−0.706 − 0.0336i)10-s + (−0.370 + 0.641i)11-s + (−0.895 − 0.239i)12-s + (0.245 + 0.245i)13-s + (−0.849 − 1.64i)15-s + (0.125 + 0.216i)16-s + (0.304 + 1.13i)17-s + (−0.445 − 1.66i)18-s + (0.119 + 0.207i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.205 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.205 + 0.978i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ -0.205 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58799 - 1.95682i\)
\(L(\frac12)\) \(\approx\) \(1.58799 - 1.95682i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (0.475 + 2.18i)T \)
7 \( 1 \)
good3 \( 1 + (-3.10 + 0.830i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (1.22 - 2.12i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.884 - 0.884i)T + 13iT^{2} \)
17 \( 1 + (-1.25 - 4.69i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.522 - 0.904i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (7.95 + 2.13i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 1.56iT - 29T^{2} \)
31 \( 1 + (-0.594 - 0.343i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.706 + 2.63i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 7.63iT - 41T^{2} \)
43 \( 1 + (-4.56 + 4.56i)T - 43iT^{2} \)
47 \( 1 + (-9.82 - 2.63i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.462 - 1.72i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.42 - 5.93i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.544 + 0.314i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.507 - 0.136i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 + (-9.73 + 2.60i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-7.06 + 4.07i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.80 + 6.80i)T + 83iT^{2} \)
89 \( 1 + (3.90 + 6.77i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.49 - 7.49i)T - 97iT^{2} \)
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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

−10.44853767593309552942487810900, −9.665066905262118612378443579947, −8.893588459392858557971246725145, −8.158788926858776445384527676329, −7.55293596894110904969611341484, −6.00690243584453165351641151820, −4.38152413207729577016141742256, −3.79119723324581311691584849711, −2.40669116161912485006532112938, −1.47233583947161588275819554250, 2.49040802559981404704752616686, 3.34806031711866203900448422241, 4.17717778282648343619504703681, 5.60405156551178899016708132319, 6.96648841215946694699127123259, 7.74342661070639568831180925010, 8.306408496686751887564443856731, 9.332824982101548684558214443775, 10.02233941246100746554504906498, 10.95476200221197937883921914482

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