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Properties

Label 2-630-9.4-c1-0-22
Degree $2$
Conductor $630$
Sign $-0.815 - 0.579i$
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  + (−0.5 − 0.866i)2-s + (−0.657 − 1.60i)3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−1.05 + 1.37i)6-s + (0.5 + 0.866i)7-s + 0.999·8-s + (−2.13 + 2.10i)9-s − 0.999·10-s + (−1.37 − 2.37i)11-s + (1.71 + 0.231i)12-s + (2.01 − 3.49i)13-s + (0.499 − 0.866i)14-s + (−1.71 − 0.231i)15-s + (−0.5 − 0.866i)16-s − 7.58·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.379 − 0.925i)3-s + (−0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s + (−0.432 + 0.559i)6-s + (0.188 + 0.327i)7-s + 0.353·8-s + (−0.711 + 0.702i)9-s − 0.316·10-s + (−0.414 − 0.717i)11-s + (0.495 + 0.0669i)12-s + (0.559 − 0.969i)13-s + (0.133 − 0.231i)14-s + (−0.443 − 0.0598i)15-s + (−0.125 − 0.216i)16-s − 1.84·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 - 0.579i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.815 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.815 - 0.579i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.815 - 0.579i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.154532 + 0.484416i\)
\(L(\frac12)\) \(\approx\) \(0.154532 + 0.484416i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (0.657 + 1.60i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good11 \( 1 + (1.37 + 2.37i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.01 + 3.49i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 7.58T + 17T^{2} \)
19 \( 1 + 4.27T + 19T^{2} \)
23 \( 1 + (-0.873 + 1.51i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.67 + 8.10i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.27 - 9.12i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.30T + 37T^{2} \)
41 \( 1 + (1.51 - 2.63i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.644 - 1.11i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.378 - 0.655i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 7.23T + 53T^{2} \)
59 \( 1 + (-0.612 + 1.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.16 + 10.6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.64 - 11.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.550T + 71T^{2} \)
73 \( 1 + 9.07T + 73T^{2} \)
79 \( 1 + (3.14 + 5.44i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.18 + 10.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 7.02T + 89T^{2} \)
97 \( 1 + (-7.79 - 13.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.36769074792833283918700252907, −8.901679425695714897363755540127, −8.524433025255262738652129295495, −7.61944969755350840433052017698, −6.40053066236301056784524066984, −5.63221090116424828678250586749, −4.45914585884115818392003196356, −2.85640863516264206578258713863, −1.83200772442889406083912047009, −0.31078450783809097514467536341, 2.12841428475819288349096045225, 3.96983267550546519952688074784, 4.62626484103366144189219218808, 5.79647597796660280981975241524, 6.63568532776518014291675571749, 7.44402725169564144686368073934, 8.860890422319954800191460617201, 9.190204554467631543778330396907, 10.27593016882491192924448128461, 10.94705287398532109140111527984

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