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Properties

Label 2-630-63.47-c1-0-0
Degree $2$
Conductor $630$
Sign $-0.901 + 0.432i$
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 + (−1.18 + 1.26i)3-s + (0.499 − 0.866i)4-s − 5-s + (0.389 − 1.68i)6-s + (2.05 − 1.66i)7-s + 0.999i·8-s + (−0.208 − 2.99i)9-s + (0.866 − 0.5i)10-s + 3.95i·11-s + (0.506 + 1.65i)12-s + (−2.82 + 1.63i)13-s + (−0.953 + 2.46i)14-s + (1.18 − 1.26i)15-s + (−0.5 − 0.866i)16-s + (0.497 + 0.861i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.682 + 0.731i)3-s + (0.249 − 0.433i)4-s − 0.447·5-s + (0.159 − 0.688i)6-s + (0.778 − 0.627i)7-s + 0.353i·8-s + (−0.0695 − 0.997i)9-s + (0.273 − 0.158i)10-s + 1.19i·11-s + (0.146 + 0.478i)12-s + (−0.784 + 0.452i)13-s + (−0.254 + 0.659i)14-s + (0.305 − 0.327i)15-s + (−0.125 − 0.216i)16-s + (0.120 + 0.208i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 + 0.432i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.901 + 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.901 + 0.432i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.901 + 0.432i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0424560 - 0.186769i\)
\(L(\frac12)\) \(\approx\) \(0.0424560 - 0.186769i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.18 - 1.26i)T \)
5 \( 1 + T \)
7 \( 1 + (-2.05 + 1.66i)T \)
good11 \( 1 - 3.95iT - 11T^{2} \)
13 \( 1 + (2.82 - 1.63i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.497 - 0.861i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.90 + 2.83i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.70iT - 23T^{2} \)
29 \( 1 + (4.10 + 2.36i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.83 + 2.79i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.05 - 8.75i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.16 + 8.94i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.10 + 8.83i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.36 + 4.10i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.82 - 5.09i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.70 - 2.95i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.45 + 3.14i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.61 - 7.98i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.74iT - 71T^{2} \)
73 \( 1 + (12.7 - 7.36i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.920 - 1.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.789 + 1.36i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.92 - 11.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.776 - 0.448i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.95783551839242859942526769129, −10.18575651577465252104577611168, −9.488813570338783833481741371785, −8.563049897748026256364565286324, −7.33965507956387850758624713930, −6.99040145179908378635498213743, −5.54552385213357130038673273905, −4.69567393366450342296656412070, −3.88276165077400000033567135196, −1.82516018646147424949308923425, 0.13581194752278408675600148424, 1.73751610796011111804761214891, 2.95666542392445010331471746152, 4.59176903699473750945409951797, 5.64668919060885285100716963243, 6.56980807999092603581673197791, 7.71599558665961247548262129652, 8.235194887207790520938672684773, 9.032305394854822588181918099575, 10.47263646618846963143085088771

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