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Properties

Label 2-630-315.299-c1-0-40
Degree $2$
Conductor $630$
Sign $-0.771 + 0.635i$
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 + (−1.13 − 1.30i)3-s + (−0.499 + 0.866i)4-s + (−2.21 − 0.314i)5-s + (0.560 − 1.63i)6-s + (2.52 + 0.790i)7-s − 0.999·8-s + (−0.406 + 2.97i)9-s + (−0.834 − 2.07i)10-s − 4.56i·11-s + (1.69 − 0.333i)12-s + (0.341 + 0.591i)13-s + (0.577 + 2.58i)14-s + (2.11 + 3.24i)15-s + (−0.5 − 0.866i)16-s + (−5.72 + 3.30i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.657 − 0.753i)3-s + (−0.249 + 0.433i)4-s + (−0.990 − 0.140i)5-s + (0.228 − 0.669i)6-s + (0.954 + 0.298i)7-s − 0.353·8-s + (−0.135 + 0.990i)9-s + (−0.263 − 0.656i)10-s − 1.37i·11-s + (0.490 − 0.0963i)12-s + (0.0946 + 0.163i)13-s + (0.154 + 0.690i)14-s + (0.544 + 0.838i)15-s + (−0.125 − 0.216i)16-s + (−1.38 + 0.802i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 + 0.635i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.771 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.771 + 0.635i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.771 + 0.635i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0951926 - 0.265323i\)
\(L(\frac12)\) \(\approx\) \(0.0951926 - 0.265323i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.13 + 1.30i)T \)
5 \( 1 + (2.21 + 0.314i)T \)
7 \( 1 + (-2.52 - 0.790i)T \)
good11 \( 1 + 4.56iT - 11T^{2} \)
13 \( 1 + (-0.341 - 0.591i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (5.72 - 3.30i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.76 + 3.90i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.50T + 23T^{2} \)
29 \( 1 + (4.33 + 2.50i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.52 + 2.60i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.37 - 0.794i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.41 + 5.91i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.05 + 1.18i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.30 - 1.90i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.983 + 1.70i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.68 + 8.11i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (12.3 - 7.15i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.28 - 3.63i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.48iT - 71T^{2} \)
73 \( 1 + (0.735 + 1.27i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.587 + 1.01i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.77 - 5.64i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.739 + 1.28i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.40 + 7.62i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.87639648007255131808337695209, −8.707322614265125912970344214925, −8.458815761339173802680294744865, −7.57219732267477044873634418605, −6.58686633892661773660629544584, −5.84479708411343089490394985704, −4.77200782142091233408220621920, −3.92237584698850362535525594192, −2.13700878717119585270734975993, −0.14269696322921726368349172299, 1.94977535240105181533968435852, 3.66443753282098518809821684085, 4.48753765527872307711467039707, 4.90564805397031666315200388906, 6.36269281045722883029107177718, 7.34979273852600177953971686659, 8.448341323691341003068247311625, 9.410290983551229033615868024146, 10.48896506660387048587122385147, 10.89595951479586724129680785076

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