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Properties

Label 1-475-475.123-r1-0-0
Degree $1$
Conductor $475$
Sign $0.243 - 0.969i$
Analytic cond. $51.0458$
Root an. cond. $51.0458$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.999 − 0.0348i)2-s + (−0.694 − 0.719i)3-s + (0.997 − 0.0697i)4-s + (−0.719 − 0.694i)6-s + (−0.866 + 0.5i)7-s + (0.994 − 0.104i)8-s + (−0.0348 + 0.999i)9-s + (0.669 + 0.743i)11-s + (−0.743 − 0.669i)12-s + (−0.469 − 0.882i)13-s + (−0.848 + 0.529i)14-s + (0.990 − 0.139i)16-s + (−0.829 − 0.559i)17-s + i·18-s + (0.961 + 0.275i)21-s + (0.694 + 0.719i)22-s + ⋯
L(s)  = 1  + (0.999 − 0.0348i)2-s + (−0.694 − 0.719i)3-s + (0.997 − 0.0697i)4-s + (−0.719 − 0.694i)6-s + (−0.866 + 0.5i)7-s + (0.994 − 0.104i)8-s + (−0.0348 + 0.999i)9-s + (0.669 + 0.743i)11-s + (−0.743 − 0.669i)12-s + (−0.469 − 0.882i)13-s + (−0.848 + 0.529i)14-s + (0.990 − 0.139i)16-s + (−0.829 − 0.559i)17-s + i·18-s + (0.961 + 0.275i)21-s + (0.694 + 0.719i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.243 - 0.969i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ 0.243 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.134613934 - 1.664577733i\)
\(L(\frac12)\) \(\approx\) \(2.134613934 - 1.664577733i\)
\(L(1)\) \(\approx\) \(1.481546564 - 0.4228088475i\)
\(L(1)\) \(\approx\) \(1.481546564 - 0.4228088475i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.999 - 0.0348i)T \)
3 \( 1 + (-0.694 - 0.719i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (0.669 + 0.743i)T \)
13 \( 1 + (-0.469 - 0.882i)T \)
17 \( 1 + (-0.829 - 0.559i)T \)
23 \( 1 + (0.927 - 0.374i)T \)
29 \( 1 + (-0.559 - 0.829i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (0.951 - 0.309i)T \)
41 \( 1 + (0.990 - 0.139i)T \)
43 \( 1 + (-0.642 + 0.766i)T \)
47 \( 1 + (0.829 - 0.559i)T \)
53 \( 1 + (-0.0697 - 0.997i)T \)
59 \( 1 + (0.615 + 0.788i)T \)
61 \( 1 + (-0.374 - 0.927i)T \)
67 \( 1 + (-0.275 - 0.961i)T \)
71 \( 1 + (-0.241 - 0.970i)T \)
73 \( 1 + (0.469 - 0.882i)T \)
79 \( 1 + (0.719 - 0.694i)T \)
83 \( 1 + (-0.406 + 0.913i)T \)
89 \( 1 + (-0.990 - 0.139i)T \)
97 \( 1 + (0.275 - 0.961i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.6431269037971456346461191712, −22.82339474125797530241916569174, −22.0123599437510350418832077300, −21.67916018069554048627338805219, −20.605051350443528111942635373190, −19.731879473973383959564437804791, −18.94116385892327688517033585682, −17.231967030996315879260070093124, −16.78319866830819444574768189757, −16.02745697765088192575760550793, −15.18260345884365149178849006844, −14.28682505123085717728474733386, −13.3369369255512359079002091933, −12.46483837003615489026889595524, −11.46824983345227099914068544599, −10.905985605348983782718326170642, −9.84691081936836892550213946976, −8.89219488497850172004568317213, −7.14387602172423402252644400390, −6.463503526807129117306174039278, −5.681248662308819588104926486168, −4.438109484832233479220641311114, −3.877514490344040591903394282564, −2.80530345184745871367609424903, −1.06809934629128585897192705894, 0.614152237248807897375166740413, 2.1079496251567117933305544415, 2.93510386285510356751128280370, 4.40510040370957328117271920449, 5.29843263484435708910758868858, 6.28902717185692421380379357830, 6.877534430535646883397914611204, 7.83388530380165637184994626093, 9.402639229091257465681629314638, 10.489322775596888425164803455967, 11.49654075854000163530985111486, 12.24876850059456457666846555647, 12.89665571564721648743669902330, 13.55281401300622184870830932360, 14.80435508814461201419765980533, 15.538758364998728529275673321323, 16.504788556229482032499516298990, 17.31646366797942460200790065172, 18.2949558040505756639841798078, 19.46819715831533763493577313106, 19.85195688148208086669787949615, 21.071841380873809321383264426512, 22.21181584793136167130655621832, 22.631156619726622772162202481008, 23.092149537017222773433614968170

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