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Properties

Label 1-53-53.9-r0-0-0
Degree $1$
Conductor $53$
Sign $0.899 - 0.436i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(53\)
Sign: $0.899 - 0.436i$
Analytic conductor: \(0.246130\)
Root analytic conductor: \(0.246130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{53} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 53,\ (0:\ ),\ 0.899 - 0.436i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8076448307 - 0.1855149870i\)
\(L(\frac12)\) \(\approx\) \(0.8076448307 - 0.1855149870i\)
\(L(1)\) \(\approx\) \(0.9266676864 - 0.1812204119i\)
\(L(1)\) \(\approx\) \(0.9266676864 - 0.1812204119i\)

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

−33.55540775718924911806819147308, −32.56137471789226203015064695366, −31.02186506299169983132123535300, −30.26754792970793713272591917800, −28.93644978762750604572521266641, −27.21377738906331541052655423860, −26.45066375225246840722465595874, −25.632442774213483890416149161886, −24.30094964201270115439136221026, −23.593203112814408402149761232173, −22.14669259552242130017459607946, −20.074911813572558815277848881386, −19.34919191762355789521407868412, −18.075466295155321623859627265949, −17.290545139597125869293049913055, −15.44560388459183683917961770888, −14.35139296038827636425656086329, −13.765226544539118223479330618525, −11.53024212981669568204062414814, −9.975975537473391308936256513819, −8.60964437938733297124803916599, −7.17878629146485088986794590735, −6.69832886060314155438188989959, −4.1860063211993476318779617113, −1.868663850388552420465481779843, 1.931210076710114181387727034839, 3.65704360884561282988051720650, 5.06140021942381380473110257170, 8.07877101203797065710926631490, 8.77545005410362853125693426099, 9.867492550909264924208078768731, 11.443111235641030983421005065803, 12.621395206105386633515083305976, 14.07380643463102561970725095840, 15.66881372311756670890156669039, 16.81854342448587552716669925783, 18.19598412457773782927188965991, 19.71130553237255527638612115556, 20.2996102326873753893655165309, 21.44594545037521651226983399738, 22.2807693630387493670183392134, 24.60682121876999673589223957279, 25.279628697158936527752420594970, 26.93512112836440146555655005709, 27.54958447253423001666866996882, 28.432374549685964144008515731614, 29.86977007816732762454777574638, 31.201917233819546781908269839635, 31.78554945719924885941508027477, 32.99944790963146838695510487919

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