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Properties

Label 2-490-35.9-c1-0-11
Degree $2$
Conductor $490$
Sign $-0.556 + 0.830i$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−1.86 + 1.23i)5-s − 0.999i·8-s + (−1.5 + 2.59i)9-s + (2.23 − 0.133i)10-s + (−1.5 − 2.59i)11-s − 5i·13-s + (−0.5 + 0.866i)16-s + (1.73 − i)17-s + (2.59 − 1.5i)18-s + (2.5 − 4.33i)19-s + (−1.99 − i)20-s + 3i·22-s + (−6.06 − 3.5i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.834 + 0.550i)5-s − 0.353i·8-s + (−0.5 + 0.866i)9-s + (0.705 − 0.0423i)10-s + (−0.452 − 0.783i)11-s − 1.38i·13-s + (−0.125 + 0.216i)16-s + (0.420 − 0.242i)17-s + (0.612 − 0.353i)18-s + (0.573 − 0.993i)19-s + (−0.447 − 0.223i)20-s + 0.639i·22-s + (−1.26 − 0.729i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.556 + 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.556 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.556 + 0.830i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ -0.556 + 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.221793 - 0.415621i\)
\(L(\frac12)\) \(\approx\) \(0.221793 - 0.415621i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (1.86 - 1.23i)T \)
7 \( 1 \)
good3 \( 1 + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5iT - 13T^{2} \)
17 \( 1 + (-1.73 + i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.06 + 3.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.866 + 0.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + (6.06 + 3.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.79 - 4.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.73 + i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (13.8 - 8i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7 + 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + (1 - 1.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12iT - 97T^{2} \)
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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

−10.63093448098266904601203181656, −10.07949134732344747248279083897, −8.611790341663077306710378505492, −8.041820099794990786829978957298, −7.38014902304940038563148187634, −6.05405266467559695997352493370, −4.89241747775404754213773638305, −3.35069069715510860054288926039, −2.61609608103738851102870849671, −0.34832747559439487807399429975, 1.58691477413178640108174232887, 3.49116095007911587172906467177, 4.60000049404850850128388095999, 5.80867994051678738572591072072, 6.85409811869460402915917281336, 7.78411371435771016112859780838, 8.502398832744738729387770225085, 9.453508158032179265214576173572, 10.07794302115675115127650664010, 11.46755781958416991414567009983

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