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Properties

Label 2-630-105.32-c1-0-14
Degree $2$
Conductor $630$
Sign $0.254 + 0.967i$
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  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.988 − 2.00i)5-s + (−2.29 − 1.31i)7-s + (0.707 + 0.707i)8-s + (−0.435 − 2.19i)10-s + (−1.05 − 0.608i)11-s + (2.42 − 2.42i)13-s + (−1.87 − 1.86i)14-s + (0.500 + 0.866i)16-s + (−0.896 − 3.34i)17-s + (7.32 − 4.22i)19-s + (0.146 − 2.23i)20-s + (−0.859 − 0.859i)22-s + (1.91 − 7.15i)23-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.433 + 0.249i)4-s + (−0.442 − 0.896i)5-s + (−0.867 − 0.497i)7-s + (0.249 + 0.249i)8-s + (−0.137 − 0.693i)10-s + (−0.317 − 0.183i)11-s + (0.672 − 0.672i)13-s + (−0.501 − 0.498i)14-s + (0.125 + 0.216i)16-s + (−0.217 − 0.811i)17-s + (1.68 − 0.970i)19-s + (0.0328 − 0.498i)20-s + (−0.183 − 0.183i)22-s + (0.399 − 1.49i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.254 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.254 + 0.967i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.254 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35931 - 1.04808i\)
\(L(\frac12)\) \(\approx\) \(1.35931 - 1.04808i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 \)
5 \( 1 + (0.988 + 2.00i)T \)
7 \( 1 + (2.29 + 1.31i)T \)
good11 \( 1 + (1.05 + 0.608i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.42 + 2.42i)T - 13iT^{2} \)
17 \( 1 + (0.896 + 3.34i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-7.32 + 4.22i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.91 + 7.15i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 7.86T + 29T^{2} \)
31 \( 1 + (3.47 - 6.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.12 - 4.19i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 5.75iT - 41T^{2} \)
43 \( 1 + (-1.25 + 1.25i)T - 43iT^{2} \)
47 \( 1 + (-7.27 - 1.94i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (8.03 - 2.15i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.0764 + 0.132i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.62 - 11.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-13.0 + 3.50i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 11.6iT - 71T^{2} \)
73 \( 1 + (2.06 + 7.69i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.09 + 0.634i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.08 - 3.08i)T + 83iT^{2} \)
89 \( 1 + (1.92 + 3.33i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.22 - 3.22i)T + 97iT^{2} \)
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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

−10.59255085345232419965087869031, −9.419604524838443699045071719599, −8.690624497934956632687414653621, −7.54648307237755963026237338134, −6.91620255299592725985726540543, −5.64045454871076988688062852204, −4.93589902193584655824462960556, −3.78687800132103155805302761044, −2.90420185569182226146647201513, −0.76278820500121784805351177603, 1.98336097332641764933963579909, 3.39256816095341205338968671381, 3.79416145735740741621466616220, 5.48232847913917083871457343280, 6.12195900535723625258761460065, 7.14655398556431172527952083941, 7.87354059054826211260799763535, 9.338192609560191683780647908797, 9.908670463062126430702833083765, 11.12289198699390004264932891841

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