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Properties

Label 1-57-57.5-r1-0-0
Degree $1$
Conductor $57$
Sign $-0.884 + 0.465i$
Analytic cond. $6.12550$
Root an. cond. $6.12550$
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  + (−0.766 + 0.642i)2-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)8-s + (0.766 + 0.642i)10-s + (0.5 + 0.866i)11-s + (−0.939 + 0.342i)13-s + (−0.173 − 0.984i)14-s + (−0.939 − 0.342i)16-s + (−0.766 + 0.642i)17-s − 20-s + (−0.939 − 0.342i)22-s + (−0.173 + 0.984i)23-s + (−0.939 + 0.342i)25-s + (0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)8-s + (0.766 + 0.642i)10-s + (0.5 + 0.866i)11-s + (−0.939 + 0.342i)13-s + (−0.173 − 0.984i)14-s + (−0.939 − 0.342i)16-s + (−0.766 + 0.642i)17-s − 20-s + (−0.939 − 0.342i)22-s + (−0.173 + 0.984i)23-s + (−0.939 + 0.342i)25-s + (0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.884 + 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.884 + 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(57\)    =    \(3 \cdot 19\)
Sign: $-0.884 + 0.465i$
Analytic conductor: \(6.12550\)
Root analytic conductor: \(6.12550\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{57} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 57,\ (1:\ ),\ -0.884 + 0.465i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1095044611 + 0.4430338957i\)
\(L(\frac12)\) \(\approx\) \(0.1095044611 + 0.4430338957i\)
\(L(1)\) \(\approx\) \(0.5269534825 + 0.2153997467i\)
\(L(1)\) \(\approx\) \(0.5269534825 + 0.2153997467i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.766 + 0.642i)T \)
5 \( 1 + (-0.173 - 0.984i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.939 + 0.342i)T \)
17 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (-0.766 - 0.642i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (-0.173 + 0.984i)T \)
59 \( 1 + (-0.766 + 0.642i)T \)
61 \( 1 + (0.173 - 0.984i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (-0.939 - 0.342i)T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (0.939 - 0.342i)T \)
97 \( 1 + (0.766 - 0.642i)T \)
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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

−32.01300507490969865060327138366, −30.67582810973950004143106875585, −29.69439203214172308582266913931, −29.14528161978252679825704014864, −27.38846001025252461652233716011, −26.74302910357759349347507729259, −25.87336330945600888532474087098, −24.40293059136121626106695570184, −22.60111156483352625615865907061, −21.99343378115916539365552502087, −20.334700950711180207522243271677, −19.45362426073574597009121020397, −18.454562424893920846373133233533, −17.20315828308083984296408877971, −16.114508525081480409972112966139, −14.39453065089934686259602307575, −13.01484166689710653408882182677, −11.45151339722021563482407412145, −10.55255962001348342080489429236, −9.37319121821096221038210223218, −7.67813679513381251277454824662, −6.63511564268361791620560801616, −3.9219705746898476961435825694, −2.636696046128937492552391647707, −0.31032801689692085046430111095, 1.85928442957834021961114120954, 4.65494985912938252198334564267, 6.05295160870750965616934356035, 7.56394257509062582759670801054, 8.99891870644067934006034512742, 9.68729486678892529765002192469, 11.653777971213873554829757196002, 12.912997588085638803202788068100, 14.75721394270663008155597160572, 15.728666569528201419100953513053, 16.860334397040323183819045380193, 17.83866014513080399483949204999, 19.36698403501722680963841741743, 20.01934332890814020309280654554, 21.724272000965295816309728551183, 23.19847558610379229459833955286, 24.45049720385606447035761003130, 25.10649725740110295995782997353, 26.29537224976903276901355696846, 27.66771384419409890629153858304, 28.349708152662496045847123114111, 29.2909453847042799453240524844, 31.23380112190428637465578719867, 32.19747121785004769244435495824, 33.16027407555831098837076938243

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