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Properties

Label 2-490-7.4-c1-0-8
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−1 + 1.73i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 1.99·6-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.499 + 0.866i)10-s + (−1.5 + 2.59i)11-s + (−0.999 − 1.73i)12-s − 5·13-s + 1.99·15-s + (−0.5 − 0.866i)16-s + (3 − 5.19i)17-s + (−0.499 + 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.577 + 0.999i)3-s + (−0.249 + 0.433i)4-s + (−0.223 − 0.387i)5-s + 0.816·6-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.158 + 0.273i)10-s + (−0.452 + 0.783i)11-s + (−0.288 − 0.499i)12-s − 1.38·13-s + 0.516·15-s + (−0.125 − 0.216i)16-s + (0.727 − 1.26i)17-s + (−0.117 + 0.204i)18-s + (−0.114 − 0.198i)19-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: $\chi_{490} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5T + 13T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.5 + 9.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + (-1.5 - 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
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 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
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

−10.44728361495533436144379240072, −9.746527302969177635810993024930, −9.220537146521944518208869142811, −7.84931969457245035496667004585, −7.10413273860930353644312497524, −5.20511365041742306298138258872, −4.90867497661907853045535280508, −3.70163549391759292414571817577, −2.23607619198832385850211950266, 0, 1.77612609997451971401408906481, 3.54234289911180153511579760492, 5.20583842442733458382234175698, 6.01904539464727457132158974265, 6.87571803693142987733167314856, 7.67529023222940634144807891197, 8.300979054420810849135242400390, 9.675812357541049788743822641253, 10.42718994449636793563267150262

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