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Properties

Label 1-475-475.22-r0-0-0
Degree $1$
Conductor $475$
Sign $0.843 + 0.536i$
Analytic cond. $2.20589$
Root an. cond. $2.20589$
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  + (−0.898 − 0.438i)2-s + (−0.529 + 0.848i)3-s + (0.615 + 0.788i)4-s + (0.848 − 0.529i)6-s + (−0.866 − 0.5i)7-s + (−0.207 − 0.978i)8-s + (−0.438 − 0.898i)9-s + (−0.104 + 0.994i)11-s + (−0.994 + 0.104i)12-s + (0.0697 − 0.997i)13-s + (0.559 + 0.829i)14-s + (−0.241 + 0.970i)16-s + (−0.139 + 0.990i)17-s + i·18-s + (0.882 − 0.469i)21-s + (0.529 − 0.848i)22-s + ⋯
L(s)  = 1  + (−0.898 − 0.438i)2-s + (−0.529 + 0.848i)3-s + (0.615 + 0.788i)4-s + (0.848 − 0.529i)6-s + (−0.866 − 0.5i)7-s + (−0.207 − 0.978i)8-s + (−0.438 − 0.898i)9-s + (−0.104 + 0.994i)11-s + (−0.994 + 0.104i)12-s + (0.0697 − 0.997i)13-s + (0.559 + 0.829i)14-s + (−0.241 + 0.970i)16-s + (−0.139 + 0.990i)17-s + i·18-s + (0.882 − 0.469i)21-s + (0.529 − 0.848i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.843 + 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.843 + 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.843 + 0.536i$
Analytic conductor: \(2.20589\)
Root analytic conductor: \(2.20589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ 0.843 + 0.536i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5339952471 + 0.1553289582i\)
\(L(\frac12)\) \(\approx\) \(0.5339952471 + 0.1553289582i\)
\(L(1)\) \(\approx\) \(0.5467680099 + 0.04889874334i\)
\(L(1)\) \(\approx\) \(0.5467680099 + 0.04889874334i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.898 - 0.438i)T \)
3 \( 1 + (-0.529 + 0.848i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (-0.104 + 0.994i)T \)
13 \( 1 + (0.0697 - 0.997i)T \)
17 \( 1 + (-0.139 + 0.990i)T \)
23 \( 1 + (0.275 - 0.961i)T \)
29 \( 1 + (0.990 - 0.139i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (0.241 - 0.970i)T \)
43 \( 1 + (-0.342 + 0.939i)T \)
47 \( 1 + (0.139 + 0.990i)T \)
53 \( 1 + (0.788 - 0.615i)T \)
59 \( 1 + (-0.719 - 0.694i)T \)
61 \( 1 + (0.961 + 0.275i)T \)
67 \( 1 + (-0.469 + 0.882i)T \)
71 \( 1 + (-0.0348 + 0.999i)T \)
73 \( 1 + (0.0697 + 0.997i)T \)
79 \( 1 + (0.848 + 0.529i)T \)
83 \( 1 + (0.743 - 0.669i)T \)
89 \( 1 + (-0.241 - 0.970i)T \)
97 \( 1 + (0.469 + 0.882i)T \)
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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

−23.6977202817258539566617951510, −23.3532544101798560161142086808, −22.13506676443444777739857455627, −21.25507430084801592996154813532, −19.70486635157203973036043261466, −19.39253931728777191705670005965, −18.41010881298033980517066004276, −18.05243282260789194594672002175, −16.6682705725043414201870783167, −16.41026455846056438278553956516, −15.46583044240079279152781193239, −14.09147260717653430685621506941, −13.44354632869384905122059694888, −12.121499975700047025096546147722, −11.4731896919622274468119494476, −10.55878452799578129422056263804, −9.320502234807835774425330926018, −8.69051658299078182238202056486, −7.50104606394681244789631435338, −6.72264556520750436529619947899, −5.99048077605366683811135528733, −5.110685063597141622515260476495, −3.08198358390741924461561451609, −1.97565251179354998861979305934, −0.648529949392135830372616884641, 0.82281427309415137081966605511, 2.55448401921500581631928217330, 3.60207006768898693673369928175, 4.51108092330127173922707648172, 6.01283471300362095670949383766, 6.85315218302629430482386725371, 8.01985359411124210219850864395, 9.08347287955086974846905667504, 10.118073321079522213648411083667, 10.3091892527935770344837175870, 11.34359609020075460476408189316, 12.51326529643084347001786608235, 12.96735669948801914585062683690, 14.75373668636880874785436807541, 15.56641145006073862897265310112, 16.30447887539404491101644043452, 17.19243593846086383691274814375, 17.67487541120648994315398378253, 18.75041810505957668062807072477, 19.88200032581884510933346027295, 20.331509691031862043178491755908, 21.184520650273858967959008056184, 22.240133478131650530607811725841, 22.72767861511338688660251872588, 23.75327915732891289240647218469

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