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Properties

Label 2-490-245.223-c1-0-1
Degree $2$
Conductor $490$
Sign $-0.826 - 0.563i$
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  + (−0.330 + 0.943i)2-s + (−0.736 − 1.17i)3-s + (−0.781 − 0.623i)4-s + (−0.815 − 2.08i)5-s + (1.35 − 0.308i)6-s + (−2.57 − 0.611i)7-s + (0.846 − 0.532i)8-s + (0.469 − 0.974i)9-s + (2.23 − 0.0823i)10-s + (−2.43 + 1.17i)11-s + (−0.155 + 1.37i)12-s + (−2.10 + 6.02i)13-s + (1.42 − 2.22i)14-s + (−1.84 + 2.49i)15-s + (0.222 + 0.974i)16-s + (−0.663 + 5.89i)17-s + ⋯
L(s)  = 1  + (−0.233 + 0.667i)2-s + (−0.425 − 0.677i)3-s + (−0.390 − 0.311i)4-s + (−0.364 − 0.931i)5-s + (0.551 − 0.125i)6-s + (−0.972 − 0.230i)7-s + (0.299 − 0.188i)8-s + (0.156 − 0.324i)9-s + (0.706 − 0.0260i)10-s + (−0.732 + 0.352i)11-s + (−0.0447 + 0.397i)12-s + (−0.584 + 1.67i)13-s + (0.381 − 0.595i)14-s + (−0.475 + 0.643i)15-s + (0.0556 + 0.243i)16-s + (−0.161 + 1.42i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 - 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.826 - 0.563i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ -0.826 - 0.563i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0437121 + 0.141770i\)
\(L(\frac12)\) \(\approx\) \(0.0437121 + 0.141770i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.330 - 0.943i)T \)
5 \( 1 + (0.815 + 2.08i)T \)
7 \( 1 + (2.57 + 0.611i)T \)
good3 \( 1 + (0.736 + 1.17i)T + (-1.30 + 2.70i)T^{2} \)
11 \( 1 + (2.43 - 1.17i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (2.10 - 6.02i)T + (-10.1 - 8.10i)T^{2} \)
17 \( 1 + (0.663 - 5.89i)T + (-16.5 - 3.78i)T^{2} \)
19 \( 1 - 4.28T + 19T^{2} \)
23 \( 1 + (-6.22 + 0.700i)T + (22.4 - 5.11i)T^{2} \)
29 \( 1 + (3.29 - 2.62i)T + (6.45 - 28.2i)T^{2} \)
31 \( 1 + 2.77iT - 31T^{2} \)
37 \( 1 + (1.98 + 0.223i)T + (36.0 + 8.23i)T^{2} \)
41 \( 1 + (11.0 + 2.51i)T + (36.9 + 17.7i)T^{2} \)
43 \( 1 + (-0.692 + 1.10i)T + (-18.6 - 38.7i)T^{2} \)
47 \( 1 + (5.59 + 1.95i)T + (36.7 + 29.3i)T^{2} \)
53 \( 1 + (-8.03 + 0.905i)T + (51.6 - 11.7i)T^{2} \)
59 \( 1 + (2.75 + 12.0i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (7.54 - 6.01i)T + (13.5 - 59.4i)T^{2} \)
67 \( 1 + (11.2 - 11.2i)T - 67iT^{2} \)
71 \( 1 + (-4.22 + 5.30i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (15.4 - 5.40i)T + (57.0 - 45.5i)T^{2} \)
79 \( 1 + 1.92iT - 79T^{2} \)
83 \( 1 + (5.92 - 2.07i)T + (64.8 - 51.7i)T^{2} \)
89 \( 1 + (-1.86 - 0.900i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (2.32 + 2.32i)T + 97iT^{2} \)
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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

−11.54091579786470003719089050671, −10.19405662054083749139249805665, −9.354315549845795590357093256802, −8.666745577238618160410413604416, −7.36004894824117642783012296134, −6.94951984083510936603008553547, −5.91682624039606797831744399338, −4.85064881548185021873534505658, −3.73424431441628218629540406350, −1.55414652255376643179626206507, 0.10225378517998147195716260484, 2.88405864222453244381197375214, 3.20539307242617803749223229132, 4.87226265207658284236946251153, 5.63344961095666710592118888382, 7.15346216391704034702587214184, 7.79014071461205775148478059435, 9.184838053667056417807812736767, 10.10727185192131138072697600456, 10.42265250944149036530553645666

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