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Properties

Label 2-630-9.4-c1-0-7
Degree $2$
Conductor $630$
Sign $0.514 - 0.857i$
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 + (1.62 + 0.606i)3-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.285 − 1.70i)6-s + (0.5 + 0.866i)7-s + 0.999·8-s + (2.26 + 1.96i)9-s + 0.999·10-s + (1.28 + 2.22i)11-s + (−1.33 + 1.10i)12-s + (−3.45 + 5.99i)13-s + (0.499 − 0.866i)14-s + (−1.33 + 1.10i)15-s + (−0.5 − 0.866i)16-s − 4.52·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.936 + 0.350i)3-s + (−0.249 + 0.433i)4-s + (−0.223 + 0.387i)5-s + (−0.116 − 0.697i)6-s + (0.188 + 0.327i)7-s + 0.353·8-s + (0.754 + 0.655i)9-s + 0.316·10-s + (0.387 + 0.671i)11-s + (−0.385 + 0.318i)12-s + (−0.959 + 1.66i)13-s + (0.133 − 0.231i)14-s + (−0.345 + 0.284i)15-s + (−0.125 − 0.216i)16-s − 1.09·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.514 - 0.857i$
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: \((2,\ 630,\ (\ :1/2),\ 0.514 - 0.857i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26044 + 0.713249i\)
\(L(\frac12)\) \(\approx\) \(1.26044 + 0.713249i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 - 0.606i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good11 \( 1 + (-1.28 - 2.22i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.45 - 5.99i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.52T + 17T^{2} \)
19 \( 1 + 5.67T + 19T^{2} \)
23 \( 1 + (-4.03 + 6.98i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.21 - 5.56i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.57 + 2.72i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.24T + 37T^{2} \)
41 \( 1 + (-4.38 + 7.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.60 - 9.70i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.143 - 0.249i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4.53T + 53T^{2} \)
59 \( 1 + (-3.55 + 6.15i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.12 - 5.40i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.50 - 9.54i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.0T + 71T^{2} \)
73 \( 1 + 1.85T + 73T^{2} \)
79 \( 1 + (2.81 + 4.87i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.24 + 3.88i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 + (6.70 + 11.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.67952161060059975304980907513, −9.779398591525323316018462244255, −8.972872544569662484342250423773, −8.545267723650309730635763860939, −7.24208153096729731825646767262, −6.66490032572331424546846017484, −4.46179509876493465374945709895, −4.35168725200362786567731039734, −2.64171364980915630447401414765, −2.03941468358774778276041679434, 0.807160995093518002325271732567, 2.49540626274668138113732062089, 3.83168734702416820479212658245, 4.92178780574405208936377907071, 6.14342286093539604243445624228, 7.16033104673138632486328592440, 7.916764990337484605502386966851, 8.536777963355961508118185131400, 9.318468012849885352996084977544, 10.22587745496201816610357998987

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