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Properties

Label 2-490-245.239-c1-0-20
Degree $2$
Conductor $490$
Sign $0.929 + 0.369i$
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  + (0.781 + 0.623i)2-s + (1.44 − 2.99i)3-s + (0.222 + 0.974i)4-s + (0.497 + 2.18i)5-s + (2.99 − 1.44i)6-s + (2.57 + 0.622i)7-s + (−0.433 + 0.900i)8-s + (−5.01 − 6.29i)9-s + (−0.970 + 2.01i)10-s + (0.511 − 0.641i)11-s + (3.24 + 0.739i)12-s + (0.149 + 0.119i)13-s + (1.62 + 2.08i)14-s + (7.24 + 1.65i)15-s + (−0.900 + 0.433i)16-s + (6.62 + 1.51i)17-s + ⋯
L(s)  = 1  + (0.552 + 0.440i)2-s + (0.832 − 1.72i)3-s + (0.111 + 0.487i)4-s + (0.222 + 0.974i)5-s + (1.22 − 0.588i)6-s + (0.971 + 0.235i)7-s + (−0.153 + 0.318i)8-s + (−1.67 − 2.09i)9-s + (−0.306 + 0.637i)10-s + (0.154 − 0.193i)11-s + (0.935 + 0.213i)12-s + (0.0414 + 0.0330i)13-s + (0.433 + 0.558i)14-s + (1.87 + 0.427i)15-s + (−0.225 + 0.108i)16-s + (1.60 + 0.366i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.929 + 0.369i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ 0.929 + 0.369i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.61156 - 0.500359i\)
\(L(\frac12)\) \(\approx\) \(2.61156 - 0.500359i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.781 - 0.623i)T \)
5 \( 1 + (-0.497 - 2.18i)T \)
7 \( 1 + (-2.57 - 0.622i)T \)
good3 \( 1 + (-1.44 + 2.99i)T + (-1.87 - 2.34i)T^{2} \)
11 \( 1 + (-0.511 + 0.641i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (-0.149 - 0.119i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-6.62 - 1.51i)T + (15.3 + 7.37i)T^{2} \)
19 \( 1 + 5.28T + 19T^{2} \)
23 \( 1 + (4.89 - 1.11i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-1.64 + 7.19i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + 1.77T + 31T^{2} \)
37 \( 1 + (-1.95 - 0.445i)T + (33.3 + 16.0i)T^{2} \)
41 \( 1 + (3.32 + 1.60i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-3.11 - 6.46i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (8.00 + 6.38i)T + (10.4 + 45.8i)T^{2} \)
53 \( 1 + (-4.25 + 0.970i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (9.28 - 4.47i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-0.643 + 2.81i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 - 9.37iT - 67T^{2} \)
71 \( 1 + (-1.73 - 7.60i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (0.624 - 0.498i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 5.95T + 79T^{2} \)
83 \( 1 + (1.26 - 1.01i)T + (18.4 - 80.9i)T^{2} \)
89 \( 1 + (2.70 + 3.39i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 13.4iT - 97T^{2} \)
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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

−11.35701360132277896239608912935, −9.959888644765622821980153465150, −8.540760279367438060050269402030, −7.969099584981928846524862268243, −7.34035333285218907400348448104, −6.31234604013460226606312448005, −5.77085513869532196880185688681, −3.83791303563886128185880682040, −2.68305733886453320441044907077, −1.72436473856503496742348061741, 1.89164652197150718779974007600, 3.35379337844754494747348359745, 4.34017825127190524409997775911, 4.89873887956647067309079178051, 5.73909738419517437762247076304, 7.83143718495303186137033557031, 8.558940721456946842593309964900, 9.374418832571665979600707903653, 10.18359651288366367507678769623, 10.77057351498682326754329338478

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