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Properties

Label 1-475-475.108-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: $\chi_{475} (108, \cdot )$
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

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

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