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Properties

Label 2-630-35.9-c1-0-2
Degree $2$
Conductor $630$
Sign $-0.758 - 0.652i$
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 + (−0.133 + 2.23i)5-s + (−1.73 + 2i)7-s − 0.999i·8-s + (1.23 − 1.86i)10-s + (1.5 + 2.59i)11-s − 5i·13-s + (2.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (−1.73 + i)17-s + (−2.5 + 4.33i)19-s + (−1.99 + 0.999i)20-s − 3i·22-s + (−6.06 − 3.5i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.0599 + 0.998i)5-s + (−0.654 + 0.755i)7-s − 0.353i·8-s + (0.389 − 0.590i)10-s + (0.452 + 0.783i)11-s − 1.38i·13-s + (0.668 − 0.231i)14-s + (−0.125 + 0.216i)16-s + (−0.420 + 0.242i)17-s + (−0.573 + 0.993i)19-s + (−0.447 + 0.223i)20-s − 0.639i·22-s + (−1.26 − 0.729i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.758 - 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.758 - 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.758 - 0.652i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.758 - 0.652i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.176941 + 0.477089i\)
\(L(\frac12)\) \(\approx\) \(0.176941 + 0.477089i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.133 - 2.23i)T \)
7 \( 1 + (1.73 - 2i)T \)
good11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5iT - 13T^{2} \)
17 \( 1 + (1.73 - i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.06 + 3.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 + (-6.06 - 3.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.79 - 4.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.73 - i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (13.8 - 8i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7 + 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + (1 - 1.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12iT - 97T^{2} \)
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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

−10.58853577348420201385062090190, −10.21696295693232661695831586077, −9.366315835593753442559603245763, −8.321473631766489335131972496420, −7.54838586190778368070410461149, −6.47491304887185274567948435795, −5.82244423395202215946973716866, −4.08517677212029222670744695388, −3.02857884017755800567062446034, −2.05113003131218847613026516857, 0.32279773276218022558764646914, 1.84720851107884458435957873793, 3.74547029634021612883503352458, 4.63856223281481133676216065331, 5.95779143696496832458662232213, 6.72586798548455458689580158667, 7.65210421784187620369172150453, 8.719410507008500627719649514933, 9.244789821962657993293880296203, 9.969114813401722763877621373408

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