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Properties

Label 2-630-63.58-c1-0-27
Degree $2$
Conductor $630$
Sign $-0.164 + 0.986i$
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  − 2-s + (0.927 − 1.46i)3-s + 4-s + (0.5 + 0.866i)5-s + (−0.927 + 1.46i)6-s + (0.832 − 2.51i)7-s − 8-s + (−1.27 − 2.71i)9-s + (−0.5 − 0.866i)10-s + (−0.189 + 0.328i)11-s + (0.927 − 1.46i)12-s + (−0.308 + 0.533i)13-s + (−0.832 + 2.51i)14-s + (1.73 + 0.0717i)15-s + 16-s + (−1.97 − 3.42i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.535 − 0.844i)3-s + 0.5·4-s + (0.223 + 0.387i)5-s + (−0.378 + 0.597i)6-s + (0.314 − 0.949i)7-s − 0.353·8-s + (−0.426 − 0.904i)9-s + (−0.158 − 0.273i)10-s + (−0.0571 + 0.0990i)11-s + (0.267 − 0.422i)12-s + (−0.0854 + 0.148i)13-s + (−0.222 + 0.671i)14-s + (0.446 + 0.0185i)15-s + 0.250·16-s + (−0.479 − 0.829i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.164 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.164 + 0.986i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.164 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.806339 - 0.951628i\)
\(L(\frac12)\) \(\approx\) \(0.806339 - 0.951628i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-0.927 + 1.46i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.832 + 2.51i)T \)
good11 \( 1 + (0.189 - 0.328i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.308 - 0.533i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.97 + 3.42i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.57 + 4.45i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.37 + 2.37i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.69 - 6.39i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.37T + 31T^{2} \)
37 \( 1 + (-3.99 + 6.91i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.63 - 6.30i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.57 + 4.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 7.44T + 47T^{2} \)
53 \( 1 + (0.998 + 1.72i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 0.231T + 59T^{2} \)
61 \( 1 + 11.5T + 61T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 + 11.2T + 71T^{2} \)
73 \( 1 + (3.58 + 6.20i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 6.03T + 79T^{2} \)
83 \( 1 + (-4.29 - 7.43i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.26 + 10.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.792 + 1.37i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.29650115154543447297472111911, −9.353065461928553816749292154443, −8.621237471591456316230137114744, −7.55053760725254900131846591316, −7.09810887883630612489227447303, −6.30715793357091292957309146008, −4.77882830622914920714839447084, −3.27053475193739409626520817576, −2.21759646319098869761016061067, −0.819694694871868081607301204702, 1.81676894652539480540176291978, 2.95050589976545202734918999200, 4.27173824107383688499740758777, 5.42004362559118066201287460643, 6.21002217255043853276575794766, 7.86571762173708211265787988863, 8.305535046366156730022940249136, 9.130094588343843150441565300148, 9.845574820146553957220701598914, 10.52526907305189468117215763824

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