Properties

Label 1-53-53.47-r0-0-0
Degree $1$
Conductor $53$
Sign $0.0898 - 0.995i$
Analytic cond. $0.246130$
Root an. cond. $0.246130$
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  + (0.568 − 0.822i)2-s + (−0.748 + 0.663i)3-s + (−0.354 − 0.935i)4-s + (0.120 − 0.992i)5-s + (0.120 + 0.992i)6-s + (0.568 − 0.822i)7-s + (−0.970 − 0.239i)8-s + (0.120 − 0.992i)9-s + (−0.748 − 0.663i)10-s + (0.885 + 0.464i)11-s + (0.885 + 0.464i)12-s + (−0.354 + 0.935i)13-s + (−0.354 − 0.935i)14-s + (0.568 + 0.822i)15-s + (−0.748 + 0.663i)16-s + (−0.970 + 0.239i)17-s + ⋯
L(s)  = 1  + (0.568 − 0.822i)2-s + (−0.748 + 0.663i)3-s + (−0.354 − 0.935i)4-s + (0.120 − 0.992i)5-s + (0.120 + 0.992i)6-s + (0.568 − 0.822i)7-s + (−0.970 − 0.239i)8-s + (0.120 − 0.992i)9-s + (−0.748 − 0.663i)10-s + (0.885 + 0.464i)11-s + (0.885 + 0.464i)12-s + (−0.354 + 0.935i)13-s + (−0.354 − 0.935i)14-s + (0.568 + 0.822i)15-s + (−0.748 + 0.663i)16-s + (−0.970 + 0.239i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0898 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0898 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(53\)
Sign: $0.0898 - 0.995i$
Analytic conductor: \(0.246130\)
Root analytic conductor: \(0.246130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{53} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 53,\ (0:\ ),\ 0.0898 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6840620450 - 0.6251438268i\)
\(L(\frac12)\) \(\approx\) \(0.6840620450 - 0.6251438268i\)
\(L(1)\) \(\approx\) \(0.9298485393 - 0.5149197180i\)
\(L(1)\) \(\approx\) \(0.9298485393 - 0.5149197180i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad53 \( 1 \)
good2 \( 1 + (0.568 - 0.822i)T \)
3 \( 1 + (-0.748 + 0.663i)T \)
5 \( 1 + (0.120 - 0.992i)T \)
7 \( 1 + (0.568 - 0.822i)T \)
11 \( 1 + (0.885 + 0.464i)T \)
13 \( 1 + (-0.354 + 0.935i)T \)
17 \( 1 + (-0.970 + 0.239i)T \)
19 \( 1 + (-0.354 + 0.935i)T \)
23 \( 1 + T \)
29 \( 1 + (0.885 - 0.464i)T \)
31 \( 1 + (0.885 - 0.464i)T \)
37 \( 1 + (-0.748 + 0.663i)T \)
41 \( 1 + (0.885 + 0.464i)T \)
43 \( 1 + (-0.748 - 0.663i)T \)
47 \( 1 + (0.120 + 0.992i)T \)
59 \( 1 + (0.120 + 0.992i)T \)
61 \( 1 + (-0.970 - 0.239i)T \)
67 \( 1 + (-0.354 - 0.935i)T \)
71 \( 1 + (-0.748 - 0.663i)T \)
73 \( 1 + (-0.970 + 0.239i)T \)
79 \( 1 + (0.568 + 0.822i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.970 + 0.239i)T \)
97 \( 1 + (0.120 - 0.992i)T \)
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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

−33.80712852206733400869168170412, −32.76766790840647752892317306900, −31.18245821052252693661915054149, −30.366061887097786236050809782812, −29.49432483336252392996623621434, −27.781655994174021581501576236052, −26.7153770056248746514368530834, −25.09244323343132838033464101381, −24.61035010893629838022049675963, −23.22939864849284858936967186781, −22.26385085009380096919582637900, −21.61594633230051699966079419184, −19.29168452677879223262602227057, −17.91242929044823670662637718304, −17.42902427756675388888030490002, −15.713675268556297948590634065982, −14.63022209630754673626045078066, −13.40238529785362593371472326562, −12.046190476969748770819900088793, −11.00001396138994479506741125315, −8.6616993347269998152844996632, −7.1492026754325983208232384759, −6.22256016249025029841318506690, −4.95036073895972449844555021601, −2.715021785663709957518028222806, 1.41049037409665203179351974939, 4.190830438108280453535043392187, 4.70810027039232146783674162679, 6.405887794373231569034573778711, 8.99840237038707639287607138684, 10.13666626876480964380864644280, 11.43669352036156958526524761721, 12.34254450265334822916515601110, 13.81462977540892684746508877218, 15.124537989161103948586988458689, 16.74749055145409988478799859121, 17.5752863411013350802647764977, 19.49653614707637917973953735521, 20.65833563763901067674154823485, 21.32951023870675571290875824647, 22.623585439226291833883195805462, 23.62721126462385551474375816218, 24.58742858978730033351256179297, 26.8407928289077100852901229271, 27.648979445921168212621597039535, 28.68246773916791553107839101777, 29.4611127601006145857392928140, 30.83679626777601749151938699240, 32.04477776110681915680584503319, 33.113163127116821660091624358647

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