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Properties

Label 2-630-315.178-c1-0-41
Degree $2$
Conductor $630$
Sign $-0.811 + 0.584i$
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  + (0.707 − 0.707i)2-s + (−0.345 − 1.69i)3-s − 1.00i·4-s + (1.87 − 1.22i)5-s + (−1.44 − 0.955i)6-s + (1.36 − 2.26i)7-s + (−0.707 − 0.707i)8-s + (−2.76 + 1.17i)9-s + (0.456 − 2.18i)10-s + (0.0834 + 0.144i)11-s + (−1.69 + 0.345i)12-s + (−2.26 − 0.607i)13-s + (−0.640 − 2.56i)14-s + (−2.72 − 2.75i)15-s − 1.00·16-s + (3.82 − 1.02i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.199 − 0.979i)3-s − 0.500i·4-s + (0.836 − 0.547i)5-s + (−0.589 − 0.390i)6-s + (0.514 − 0.857i)7-s + (−0.250 − 0.250i)8-s + (−0.920 + 0.391i)9-s + (0.144 − 0.692i)10-s + (0.0251 + 0.0436i)11-s + (−0.489 + 0.0997i)12-s + (−0.629 − 0.168i)13-s + (−0.171 − 0.686i)14-s + (−0.703 − 0.710i)15-s − 0.250·16-s + (0.926 − 0.248i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.811 + 0.584i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.811 + 0.584i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.626259 - 1.93892i\)
\(L(\frac12)\) \(\approx\) \(0.626259 - 1.93892i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (0.345 + 1.69i)T \)
5 \( 1 + (-1.87 + 1.22i)T \)
7 \( 1 + (-1.36 + 2.26i)T \)
good11 \( 1 + (-0.0834 - 0.144i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.26 + 0.607i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-3.82 + 1.02i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.72 - 4.72i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.65 - 0.444i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-0.515 - 0.297i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.29iT - 31T^{2} \)
37 \( 1 + (3.70 + 0.993i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (9.70 - 5.60i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-9.11 + 2.44i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (0.553 + 0.553i)T + 47iT^{2} \)
53 \( 1 + (6.98 - 1.87i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 - 1.05T + 59T^{2} \)
61 \( 1 + 5.61iT - 61T^{2} \)
67 \( 1 + (-8.91 + 8.91i)T - 67iT^{2} \)
71 \( 1 - 9.49T + 71T^{2} \)
73 \( 1 + (-1.56 - 5.82i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 - 8.67iT - 79T^{2} \)
83 \( 1 + (-1.70 - 6.35i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (7.25 + 12.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-13.8 + 3.71i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.29149589115606187431544875324, −9.658208915526948072899407666561, −8.353332434512831005745568498864, −7.55982578286804874812961250905, −6.55761284317448620967439145771, −5.51276862117103241061899500845, −4.90630620145440493069815420995, −3.39390471778796238150242434386, −1.97519991511814724332906738828, −1.04650931717068126949269715249, 2.36268123371322938579619334083, 3.37768593570358217473857944276, 4.76208405480571098789619918766, 5.43765149150656235203975879888, 6.13907696246054267707167037840, 7.25723694753496966712539307393, 8.420564468220607185265468767579, 9.319890135507566290320999385575, 9.956625008758171316146544463693, 10.97536784365457805800561565102

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