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Properties

Label 2-490-7.2-c1-0-0
Degree $2$
Conductor $490$
Sign $-0.991 - 0.126i$
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  + (−0.5 + 0.866i)2-s + (1.5 + 2.59i)3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s − 3·6-s + 0.999·8-s + (−3 + 5.19i)9-s + (−0.499 − 0.866i)10-s + (1 + 1.73i)11-s + (1.50 − 2.59i)12-s − 3·15-s + (−0.5 + 0.866i)16-s + (−2 − 3.46i)17-s + (−3 − 5.19i)18-s + (−3 + 5.19i)19-s + 0.999·20-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.866 + 1.49i)3-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s − 1.22·6-s + 0.353·8-s + (−1 + 1.73i)9-s + (−0.158 − 0.273i)10-s + (0.301 + 0.522i)11-s + (0.433 − 0.749i)12-s − 0.774·15-s + (−0.125 + 0.216i)16-s + (−0.485 − 0.840i)17-s + (−0.707 − 1.22i)18-s + (−0.688 + 1.19i)19-s + 0.223·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.991 - 0.126i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0843398 + 1.32902i\)
\(L(\frac12)\) \(\approx\) \(0.0843398 + 1.32902i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 \)
good3 \( 1 + (-1.5 - 2.59i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 - 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 7T + 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5 - 8.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.5 - 7.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (2 + 3.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5 - 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7T + 83T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−10.97665083703086790483412661559, −10.16781257917926660068681781130, −9.624615551705869284953254924198, −8.756988663527870880694039205949, −8.019809309138138701606525226178, −6.98978401085185422934519850516, −5.70808504593567739297744596633, −4.53046448399197474587524858813, −3.82586841758677743731874991009, −2.45281983597269265673228582507, 0.839114570558405706275007986138, 2.11386345682440123814231581710, 3.13625436568813031171355036015, 4.47020055782771344255608591463, 6.26317338191743178486844825574, 6.99391885917245285038671016395, 8.211240608656270383576244000224, 8.493255312495664521477529628378, 9.335723948153828938297310453389, 10.67301765985316947394187938296

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