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Properties

Label 1-57-57.35-r1-0-0
Degree $1$
Conductor $57$
Sign $0.999 + 0.0151i$
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  + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (0.939 + 0.342i)5-s + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)8-s + (0.173 − 0.984i)10-s + (0.5 + 0.866i)11-s + (0.766 + 0.642i)13-s + (0.939 + 0.342i)14-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s − 20-s + (0.766 − 0.642i)22-s + (0.939 − 0.342i)23-s + (0.766 + 0.642i)25-s + (0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (0.939 + 0.342i)5-s + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)8-s + (0.173 − 0.984i)10-s + (0.5 + 0.866i)11-s + (0.766 + 0.642i)13-s + (0.939 + 0.342i)14-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s − 20-s + (0.766 − 0.642i)22-s + (0.939 − 0.342i)23-s + (0.766 + 0.642i)25-s + (0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(57\)    =    \(3 \cdot 19\)
Sign: $0.999 + 0.0151i$
Analytic conductor: \(6.12550\)
Root analytic conductor: \(6.12550\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{57} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 57,\ (1:\ ),\ 0.999 + 0.0151i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.438328726 + 0.01089764917i\)
\(L(\frac12)\) \(\approx\) \(1.438328726 + 0.01089764917i\)
\(L(1)\) \(\approx\) \(1.056332032 - 0.1697983833i\)
\(L(1)\) \(\approx\) \(1.056332032 - 0.1697983833i\)

Euler product

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

−32.756474701191764601408587491964, −32.131613564403584392147871681978, −30.39950288988581010364628682070, −29.20976888711920557538440809454, −28.0102424924692220690935838896, −26.74361626773467840683574424227, −25.78533576134527847279467534831, −24.856275206490450433515142612425, −23.74641017437518533446881838136, −22.60291968953533876925028261942, −21.4044707426115707506559587329, −19.839203742094336985089193081331, −18.49430894156476838854440210512, −17.14899935963971660565252067354, −16.6209229133186701085551850030, −15.11291029358574515071581259462, −13.64839716993729863394932476929, −13.130558724729049064715069750335, −10.67119098636569560365818706213, −9.46573554869979908849530477166, −8.251587784051151956152731854231, −6.59151251144078426528174596124, −5.63504687335844902771220230213, −3.83832825485390253666575249303, −0.95101056403131951767863433883, 1.73108246790774738861186715602, 3.08517801606368143935872148469, 5.01415196698658971154043341456, 6.69441886743346082698346284191, 8.94298807483856681968000171894, 9.6406803890981566646994746978, 11.06941780498358394938004468283, 12.36701533745837223528709136617, 13.486869365073046482284776040744, 14.72963836619144120970944991431, 16.59628666612786152713963528191, 18.02379249703795250006906628325, 18.64512328839376353216359664725, 20.07658821144540179319841150472, 21.25190372014157266286543888519, 22.13206114746657424619080604648, 23.08478510133871479740143565895, 25.11830670689319748407914466466, 25.87502839865910820246859556745, 27.252157974144879045106555386784, 28.54206417293490053247322375998, 29.07343945464903425021587227782, 30.373823187751449678654526608025, 31.24333685594390194957138623443, 32.51503171715948983447525413322

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