Properties

Degree 1
Conductor 53
Sign $0.0898 + 0.995i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(53\)
\( \varepsilon \)  =  $0.0898 + 0.995i$
motivic weight  =  \(0\)
character  :  $\chi_{53} (44, \cdot )$
Sato-Tate  :  $\mu(13)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 53,\ (0:\ ),\ 0.0898 + 0.995i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.6840620450 + 0.6251438268i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.6840620450 + 0.6251438268i\)
\(L(\chi,1)\)  \(\approx\)  \(0.9298485393 + 0.5149197180i\)
\(L(1,\chi)\)  \(\approx\)  \(0.9298485393 + 0.5149197180i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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