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Properties

Label 2-630-63.59-c1-0-29
Degree $2$
Conductor $630$
Sign $0.592 + 0.805i$
Analytic cond. $5.03057$
Root an. cond. $2.24289$
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 + (0.866 − 1.5i)3-s + (0.499 + 0.866i)4-s − 5-s + (1.5 − 0.866i)6-s + (0.5 − 2.59i)7-s + 0.999i·8-s + (−1.5 − 2.59i)9-s + (−0.866 − 0.5i)10-s − 1.26i·11-s + 1.73·12-s + (3 + 1.73i)13-s + (1.73 − 2i)14-s + (−0.866 + 1.5i)15-s + (−0.5 + 0.866i)16-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.499 − 0.866i)3-s + (0.249 + 0.433i)4-s − 0.447·5-s + (0.612 − 0.353i)6-s + (0.188 − 0.981i)7-s + 0.353i·8-s + (−0.5 − 0.866i)9-s + (−0.273 − 0.158i)10-s − 0.382i·11-s + 0.499·12-s + (0.832 + 0.480i)13-s + (0.462 − 0.534i)14-s + (−0.223 + 0.387i)15-s + (−0.125 + 0.216i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.592 + 0.805i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.592 + 0.805i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05919 - 1.04087i\)
\(L(\frac12)\) \(\approx\) \(2.05919 - 1.04087i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.866 + 1.5i)T \)
5 \( 1 + T \)
7 \( 1 + (-0.5 + 2.59i)T \)
good11 \( 1 + 1.26iT - 11T^{2} \)
13 \( 1 + (-3 - 1.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.09 + 2.36i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 9.46iT - 23T^{2} \)
29 \( 1 + (0.401 - 0.232i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.90 - 1.09i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.09 - 3.63i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.59 - 6.23i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.5 - 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.29 - 5.36i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.09 + 7.09i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.803 - 0.464i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.26iT - 71T^{2} \)
73 \( 1 + (-6 - 3.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.29 + 14.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.401 - 0.696i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-8.19 - 14.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.3 - 7.73i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.77566584053855227188460379342, −9.370768133236488358965954187393, −8.346214960009910801152408852764, −7.79607652137264196970515844042, −6.80679567220131257097164024274, −6.29352844880963334019350224388, −4.79488583006067507892290530226, −3.80349759170134503101220163200, −2.81480228706294801346306141533, −1.08682525589808161607387736132, 1.96249970031881433156276035156, 3.29199062693896075769752840866, 3.90010254820835112747678771613, 5.29368411791166463522667467373, 5.64435730066853738584944188419, 7.30061577312048384882896132546, 8.236078008515756389472098545156, 9.128236326171480864929142329422, 9.867590307821884872114353915764, 10.82552506130390300811292958093

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