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Properties

Label 2-630-63.59-c1-0-12
Degree $2$
Conductor $630$
Sign $0.973 - 0.229i$
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  + (0.866 + 0.5i)2-s + (−1.22 − 1.22i)3-s + (0.499 + 0.866i)4-s − 5-s + (−0.443 − 1.67i)6-s + (2.64 + 0.169i)7-s + 0.999i·8-s + (−0.0155 + 2.99i)9-s + (−0.866 − 0.5i)10-s + 0.207i·11-s + (0.452 − 1.67i)12-s + (1.22 + 0.707i)13-s + (2.20 + 1.46i)14-s + (1.22 + 1.22i)15-s + (−0.5 + 0.866i)16-s + (0.680 − 1.17i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.705 − 0.708i)3-s + (0.249 + 0.433i)4-s − 0.447·5-s + (−0.181 − 0.683i)6-s + (0.997 + 0.0641i)7-s + 0.353i·8-s + (−0.00516 + 0.999i)9-s + (−0.273 − 0.158i)10-s + 0.0626i·11-s + (0.130 − 0.482i)12-s + (0.339 + 0.196i)13-s + (0.588 + 0.392i)14-s + (0.315 + 0.317i)15-s + (−0.125 + 0.216i)16-s + (0.165 − 0.285i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.973 - 0.229i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.973 - 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76822 + 0.205968i\)
\(L(\frac12)\) \(\approx\) \(1.76822 + 0.205968i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 + T \)
7 \( 1 + (-2.64 - 0.169i)T \)
good11 \( 1 - 0.207iT - 11T^{2} \)
13 \( 1 + (-1.22 - 0.707i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.680 + 1.17i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.73 + 3.30i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 6.39iT - 23T^{2} \)
29 \( 1 + (-6.11 + 3.53i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.10 + 2.37i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.32 - 2.30i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.41 - 2.44i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.06 - 1.84i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.0573 - 0.0993i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.40 + 4.85i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.34 - 7.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.85 + 1.06i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.0235 + 0.0407i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.45iT - 71T^{2} \)
73 \( 1 + (11.5 + 6.68i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.97 - 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.06 + 13.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.40 + 2.44i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.39 + 0.803i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.14489306168793710514291423851, −9.867217429304536041151472084286, −8.497333007816474565722379669798, −7.71568923818544979862245115445, −7.12185528842530071708842988684, −6.04086982153802643747756652608, −5.15046571234168559435662086201, −4.41136007467401089424871316170, −2.86456833167719857601032116591, −1.28932493440538180621631624753, 1.15422805191874139373068366268, 3.06692470789502806593177874355, 4.14627005503824215432420210274, 4.90096573122683405557805918124, 5.72915377655300769020148083487, 6.76797177181280140381639121034, 7.967783460811675353853113643545, 8.887129422588452494615631950323, 10.14853209818691014438585515332, 10.63634549881443640709123833242

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