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Properties

Label 2-490-35.4-c1-0-12
Degree $2$
Conductor $490$
Sign $0.384 + 0.923i$
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 + (1.48 − 1.67i)5-s + 0.999i·8-s + (−1.5 − 2.59i)9-s + (−0.448 + 2.19i)10-s + (2 − 3.46i)11-s + 4.24i·13-s + (−0.5 − 0.866i)16-s + (−3.67 − 2.12i)17-s + (2.59 + 1.5i)18-s + (−2.82 − 4.89i)19-s + (−0.707 − 2.12i)20-s + 3.99i·22-s + (−0.598 − 4.96i)25-s + (−2.12 − 3.67i)26-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.663 − 0.748i)5-s + 0.353i·8-s + (−0.5 − 0.866i)9-s + (−0.141 + 0.692i)10-s + (0.603 − 1.04i)11-s + 1.17i·13-s + (−0.125 − 0.216i)16-s + (−0.891 − 0.514i)17-s + (0.612 + 0.353i)18-s + (−0.648 − 1.12i)19-s + (−0.158 − 0.474i)20-s + 0.852i·22-s + (−0.119 − 0.992i)25-s + (−0.416 − 0.720i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.384 + 0.923i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ 0.384 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.856747 - 0.571421i\)
\(L(\frac12)\) \(\approx\) \(0.856747 - 0.571421i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 + 1.67i)T \)
7 \( 1 \)
good3 \( 1 + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.24iT - 13T^{2} \)
17 \( 1 + (3.67 + 2.12i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.82 + 4.89i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (2.82 - 4.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.19 + 3i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.41T + 41T^{2} \)
43 \( 1 + 12iT - 43T^{2} \)
47 \( 1 + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-10.3 - 6i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.65 + 9.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.53 - 6.12i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.3 + 6i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (-3.67 - 2.12i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.9iT - 83T^{2} \)
89 \( 1 + (-2.12 - 3.67i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.24iT - 97T^{2} \)
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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

−10.79196698774209708067991739164, −9.492924757782086043864086324783, −8.908151022074489562386960957149, −8.601110976629843134000374342620, −6.88855535489482049312375509870, −6.37037403514766418830205341378, −5.31386649484653730912403501380, −4.08467674449296409002398588219, −2.37090541117440441318954254126, −0.76220086117549775617157546450, 1.84844258356382484747429837283, 2.79173113980320049660532182227, 4.24807527817287375369643201222, 5.69341275887557825447193770535, 6.58851643816266308161431668172, 7.66917839999462233189733964939, 8.422247769748360655522971248473, 9.586892172178479218445676642873, 10.27861576006669446163233013971, 10.86295572526166346269242938717

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