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Properties

Label 2-630-63.47-c1-0-16
Degree $2$
Conductor $630$
Sign $0.934 + 0.356i$
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 + (−1.15 + 1.29i)3-s + (0.499 − 0.866i)4-s + 5-s + (0.353 − 1.69i)6-s + (2.60 + 0.461i)7-s + 0.999i·8-s + (−0.335 − 2.98i)9-s + (−0.866 + 0.5i)10-s − 5.17i·11-s + (0.541 + 1.64i)12-s + (0.604 − 0.349i)13-s + (−2.48 + 0.903i)14-s + (−1.15 + 1.29i)15-s + (−0.5 − 0.866i)16-s + (−1.77 − 3.07i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.666 + 0.745i)3-s + (0.249 − 0.433i)4-s + 0.447·5-s + (0.144 − 0.692i)6-s + (0.984 + 0.174i)7-s + 0.353i·8-s + (−0.111 − 0.993i)9-s + (−0.273 + 0.158i)10-s − 1.56i·11-s + (0.156 + 0.474i)12-s + (0.167 − 0.0968i)13-s + (−0.664 + 0.241i)14-s + (−0.298 + 0.333i)15-s + (−0.125 − 0.216i)16-s + (−0.430 − 0.745i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.934 + 0.356i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.934 + 0.356i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.897136 - 0.165245i\)
\(L(\frac12)\) \(\approx\) \(0.897136 - 0.165245i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.15 - 1.29i)T \)
5 \( 1 - T \)
7 \( 1 + (-2.60 - 0.461i)T \)
good11 \( 1 + 5.17iT - 11T^{2} \)
13 \( 1 + (-0.604 + 0.349i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.77 + 3.07i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.54 + 2.62i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.75iT - 23T^{2} \)
29 \( 1 + (4.60 + 2.65i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.81 - 2.77i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.06 + 8.77i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.18 + 5.52i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.94 - 10.3i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.08 - 7.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.82 + 5.09i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.679 - 1.17i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.28 + 3.05i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.21 + 3.82i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.76iT - 71T^{2} \)
73 \( 1 + (3.19 - 1.84i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.81 - 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.80 - 13.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.57 + 4.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-15.5 - 9.00i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.79366588350160428283741453126, −9.613198939599907183094099333963, −8.779344709681736467647277654404, −8.247947490864548372964339650372, −6.80409784534026841011606275918, −5.99582266062224355345388912713, −5.22342658825373627022978796436, −4.21441447388559845984657548087, −2.56050516586769730899173571619, −0.69317309493941152292689455596, 1.54477453360663689123926194839, 2.12173334113316940270540412824, 4.17096593408388289705328870489, 5.19259994235605263013334030897, 6.33935938341543702291014246336, 7.21391748961962009909966444797, 7.962756098408274042975655499072, 8.824010729128065838959496061063, 10.11181726155541137946939942749, 10.50344298579275996469281626560

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