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Properties

Label 2-630-105.59-c1-0-1
Degree $2$
Conductor $630$
Sign $-0.921 + 0.387i$
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 + (−0.499 + 0.866i)4-s + (−0.442 + 2.19i)5-s + (−1.63 + 2.07i)7-s − 0.999·8-s + (−2.11 + 0.712i)10-s + (−5.48 − 3.16i)11-s − 1.05·13-s + (−2.61 − 0.381i)14-s + (−0.5 − 0.866i)16-s + (−4.01 − 2.31i)17-s + (5.35 − 3.08i)19-s + (−1.67 − 1.47i)20-s − 6.33i·22-s + (3.52 + 6.10i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.197 + 0.980i)5-s + (−0.619 + 0.784i)7-s − 0.353·8-s + (−0.670 + 0.225i)10-s + (−1.65 − 0.954i)11-s − 0.292·13-s + (−0.699 − 0.101i)14-s + (−0.125 − 0.216i)16-s + (−0.973 − 0.562i)17-s + (1.22 − 0.708i)19-s + (−0.374 − 0.330i)20-s − 1.34i·22-s + (0.734 + 1.27i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.921 + 0.387i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.921 + 0.387i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.135958 - 0.673428i\)
\(L(\frac12)\) \(\approx\) \(0.135958 - 0.673428i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (0.442 - 2.19i)T \)
7 \( 1 + (1.63 - 2.07i)T \)
good11 \( 1 + (5.48 + 3.16i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.05T + 13T^{2} \)
17 \( 1 + (4.01 + 2.31i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.35 + 3.08i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.52 - 6.10i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.98iT - 29T^{2} \)
31 \( 1 + (-5.46 - 3.15i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.01 - 1.73i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.97T + 41T^{2} \)
43 \( 1 - 2.58iT - 43T^{2} \)
47 \( 1 + (7.06 - 4.07i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.43 - 7.68i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.452 + 0.783i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.81 - 5.08i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.51 - 4.91i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.4iT - 71T^{2} \)
73 \( 1 + (4.18 - 7.24i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.73 - 4.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.6iT - 83T^{2} \)
89 \( 1 + (-1.91 - 3.32i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.87T + 97T^{2} \)
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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

−11.21795974662887345979678559369, −10.19815679176629398225316333468, −9.270701449785801287900729376465, −8.316740738088847358540119631561, −7.37989865214669701932798911093, −6.68531716966110937556836388516, −5.63473028894958420137887318367, −4.94626619262588946775640056161, −3.11215605816777073143599298849, −2.86471314497414017182628094309, 0.31279327404382499682583066022, 2.03901038909300554887798524311, 3.37748996264880651464872027817, 4.58822173372607094336998376331, 5.08250533834675368684353065087, 6.42451340290515384572970456896, 7.57121350113388366068173574898, 8.352884292563820884544093116830, 9.556630543776117487495331620652, 10.12762555423421271052444006917

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