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Properties

Label 2-630-21.17-c1-0-5
Degree $2$
Conductor $630$
Sign $0.286 + 0.958i$
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 + (0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (2.63 − 0.189i)7-s − 0.999i·8-s + (−0.866 − 0.499i)10-s + (−1.32 − 0.766i)11-s − 1.48i·13-s + (2.19 − 1.48i)14-s + (−0.5 − 0.866i)16-s + (1.21 − 2.10i)17-s + (4.21 − 2.43i)19-s − 0.999·20-s − 1.53·22-s + (0.232 − 0.133i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.223 − 0.387i)5-s + (0.997 − 0.0716i)7-s − 0.353i·8-s + (−0.273 − 0.158i)10-s + (−0.400 − 0.230i)11-s − 0.411i·13-s + (0.585 − 0.396i)14-s + (−0.125 − 0.216i)16-s + (0.294 − 0.510i)17-s + (0.966 − 0.557i)19-s − 0.223·20-s − 0.326·22-s + (0.0483 − 0.0279i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.286 + 0.958i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.286 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74334 - 1.29866i\)
\(L(\frac12)\) \(\approx\) \(1.74334 - 1.29866i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-2.63 + 0.189i)T \)
good11 \( 1 + (1.32 + 0.766i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.48iT - 13T^{2} \)
17 \( 1 + (-1.21 + 2.10i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.21 + 2.43i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.232 + 0.133i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.898iT - 29T^{2} \)
31 \( 1 + (0.717 + 0.414i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.74 - 4.74i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.76T + 41T^{2} \)
43 \( 1 - 1.86T + 43T^{2} \)
47 \( 1 + (3.72 + 6.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.00 - 1.73i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.12 + 5.41i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.73 + 3.31i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.01 - 13.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.7iT - 71T^{2} \)
73 \( 1 + (-0.297 - 0.171i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.22 - 9.04i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.45T + 83T^{2} \)
89 \( 1 + (-7.98 - 13.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 14.9iT - 97T^{2} \)
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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

−10.57677525785290901929646617098, −9.718235021005033634366369411681, −8.596673105040372923174585925311, −7.80555997342465037260448761587, −6.85567580020172186693159069060, −5.38505662434304207812700413544, −5.02832577781539344154942033632, −3.80677870948419169670818356803, −2.61420442223824258818250812734, −1.10071313281472757115748315523, 1.83591520612830814397878669756, 3.23488926618545397573917776564, 4.33994796659624762056572775222, 5.25570973637353364567198369334, 6.17046166667693508237427802262, 7.35769580453227931802676905163, 7.85741490521215889305291080640, 8.843210972803179857484143098488, 10.04809565006938576546434383804, 10.92890371037024769509715066987

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