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Properties

Label 2-630-9.4-c1-0-17
Degree $2$
Conductor $630$
Sign $0.407 + 0.913i$
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 + (1.71 − 0.211i)3-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−1.04 − 1.38i)6-s + (−0.5 − 0.866i)7-s + 0.999·8-s + (2.91 − 0.728i)9-s + 0.999·10-s + (−1.04 − 1.80i)11-s + (−0.675 + 1.59i)12-s + (1.54 − 2.67i)13-s + (−0.499 + 0.866i)14-s + (−0.675 + 1.59i)15-s + (−0.5 − 0.866i)16-s + 1.64·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.992 − 0.122i)3-s + (−0.249 + 0.433i)4-s + (−0.223 + 0.387i)5-s + (−0.425 − 0.564i)6-s + (−0.188 − 0.327i)7-s + 0.353·8-s + (0.970 − 0.242i)9-s + 0.316·10-s + (−0.314 − 0.544i)11-s + (−0.195 + 0.460i)12-s + (0.427 − 0.741i)13-s + (−0.133 + 0.231i)14-s + (−0.174 + 0.411i)15-s + (−0.125 − 0.216i)16-s + 0.399·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.407 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.407 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.407 + 0.913i$
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: \((2,\ 630,\ (\ :1/2),\ 0.407 + 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39860 - 0.907272i\)
\(L(\frac12)\) \(\approx\) \(1.39860 - 0.907272i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 + 0.211i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good11 \( 1 + (1.04 + 1.80i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.54 + 2.67i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.64T + 17T^{2} \)
19 \( 1 - 4.52T + 19T^{2} \)
23 \( 1 + (-1.19 + 2.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.895 - 1.55i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.43 + 4.22i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.90T + 37T^{2} \)
41 \( 1 + (3.74 - 6.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.413 + 0.716i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.64 + 2.85i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 3.73T + 53T^{2} \)
59 \( 1 + (5.17 - 8.96i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.21 + 2.11i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.70 - 9.87i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.6T + 71T^{2} \)
73 \( 1 - 1.70T + 73T^{2} \)
79 \( 1 + (-0.351 - 0.609i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.73 + 6.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.34T + 89T^{2} \)
97 \( 1 + (-3.93 - 6.82i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.27486353349283521976105673167, −9.700264481197891596266176537424, −8.670509560609899590597654534923, −7.930648617915324722266404351882, −7.28721762281056321659051827415, −6.01537180678640738105815202518, −4.48576990352188606176268523390, −3.32649012175288685656479027039, −2.78700241396403781626685291726, −1.08460016557594220283305499380, 1.52979587573625713879998115800, 3.03751856035070876076257029286, 4.27310847496562127988847521495, 5.21676780385973644524711758009, 6.48441459582091330734917048040, 7.48846811818148738494840178797, 8.078686554119714666859491751749, 9.077537322894632697842039194078, 9.508974515271850290955870788310, 10.40221380394177598897547736541

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