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Properties

Label 2-490-245.237-c1-0-7
Degree $2$
Conductor $490$
Sign $0.513 - 0.857i$
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  + (−0.111 + 0.993i)2-s + (−0.753 − 0.263i)3-s + (−0.974 − 0.222i)4-s + (−2.23 + 0.0551i)5-s + (0.346 − 0.719i)6-s + (2.24 − 1.39i)7-s + (0.330 − 0.943i)8-s + (−1.84 − 1.47i)9-s + (0.195 − 2.22i)10-s + (2.32 + 2.90i)11-s + (0.676 + 0.424i)12-s + (−0.342 + 3.03i)13-s + (1.13 + 2.39i)14-s + (1.69 + 0.548i)15-s + (0.900 + 0.433i)16-s + (3.98 + 2.50i)17-s + ⋯
L(s)  = 1  + (−0.0791 + 0.702i)2-s + (−0.435 − 0.152i)3-s + (−0.487 − 0.111i)4-s + (−0.999 + 0.0246i)5-s + (0.141 − 0.293i)6-s + (0.849 − 0.527i)7-s + (0.116 − 0.333i)8-s + (−0.615 − 0.490i)9-s + (0.0618 − 0.704i)10-s + (0.699 + 0.877i)11-s + (0.195 + 0.122i)12-s + (−0.0949 + 0.842i)13-s + (0.303 + 0.638i)14-s + (0.438 + 0.141i)15-s + (0.225 + 0.108i)16-s + (0.966 + 0.607i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 - 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.513 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.513 - 0.857i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ 0.513 - 0.857i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.883384 + 0.500685i\)
\(L(\frac12)\) \(\approx\) \(0.883384 + 0.500685i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.111 - 0.993i)T \)
5 \( 1 + (2.23 - 0.0551i)T \)
7 \( 1 + (-2.24 + 1.39i)T \)
good3 \( 1 + (0.753 + 0.263i)T + (2.34 + 1.87i)T^{2} \)
11 \( 1 + (-2.32 - 2.90i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (0.342 - 3.03i)T + (-12.6 - 2.89i)T^{2} \)
17 \( 1 + (-3.98 - 2.50i)T + (7.37 + 15.3i)T^{2} \)
19 \( 1 - 8.50T + 19T^{2} \)
23 \( 1 + (2.65 + 4.21i)T + (-9.97 + 20.7i)T^{2} \)
29 \( 1 + (2.00 - 0.457i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 - 8.37iT - 31T^{2} \)
37 \( 1 + (-0.571 + 0.909i)T + (-16.0 - 33.3i)T^{2} \)
41 \( 1 + (-1.31 - 2.72i)T + (-25.5 + 32.0i)T^{2} \)
43 \( 1 + (-7.85 + 2.74i)T + (33.6 - 26.8i)T^{2} \)
47 \( 1 + (9.01 + 1.01i)T + (45.8 + 10.4i)T^{2} \)
53 \( 1 + (-2.26 - 3.61i)T + (-22.9 + 47.7i)T^{2} \)
59 \( 1 + (-10.5 - 5.06i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (-11.6 + 2.66i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 + (9.61 + 9.61i)T + 67iT^{2} \)
71 \( 1 + (-0.411 + 1.80i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (4.01 - 0.452i)T + (71.1 - 16.2i)T^{2} \)
79 \( 1 - 9.67iT - 79T^{2} \)
83 \( 1 + (-0.184 + 0.0208i)T + (80.9 - 18.4i)T^{2} \)
89 \( 1 + (-6.35 + 7.96i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + (0.159 - 0.159i)T - 97iT^{2} \)
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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

−11.32247490639530463275004430772, −10.19294351209714507822437572210, −9.137346159074342789193543103316, −8.221990394025531559706794625802, −7.35344576574078277870981255131, −6.75988863258383254617087242310, −5.47471105943932953267885793400, −4.50843816777769697282114536802, −3.54890010430144173802010644628, −1.16125663078333597519950447389, 0.892482802199530110809395213109, 2.83707052718135230530619745281, 3.82098173703585044658339950698, 5.19988292964915236537440751540, 5.68239573726895917176738902055, 7.63995738818985603927509779129, 8.023105607634692135199722828355, 9.088161511692253424862941744381, 10.06377107900853290447158701832, 11.29589228354220872612498778655

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