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Properties

Label 1-57-57.56-r0-0-0
Degree $1$
Conductor $57$
Sign $1$
Analytic cond. $0.264706$
Root an. cond. $0.264706$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s − 11-s − 13-s + 14-s + 16-s − 17-s − 20-s − 22-s − 23-s + 25-s − 26-s + 28-s + 29-s − 31-s + 32-s − 34-s − 35-s − 37-s − 40-s + 41-s + 43-s − 44-s + ⋯
L(s)  = 1  + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s − 11-s − 13-s + 14-s + 16-s − 17-s − 20-s − 22-s − 23-s + 25-s − 26-s + 28-s + 29-s − 31-s + 32-s − 34-s − 35-s − 37-s − 40-s + 41-s + 43-s − 44-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(57\)    =    \(3 \cdot 19\)
Sign: $1$
Analytic conductor: \(0.264706\)
Root analytic conductor: \(0.264706\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{57} (56, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 57,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.385011232\)
\(L(\frac12)\) \(\approx\) \(1.385011232\)
\(L(1)\) \(\approx\) \(1.512726166\)
\(L(1)\) \(\approx\) \(1.512726166\)

Euler product

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

−32.8182887332261932697096796825, −31.40445266752256121352949573385, −31.128217180029262240231189064165, −29.8578836602481893856332621913, −28.62351137889036648079428283193, −27.3242642506575048000894215764, −26.138076418826237379980199526152, −24.40192557124825977782532324401, −23.96150603262041493797356734403, −22.76715949979014002691448233615, −21.57822376673138383783744370346, −20.43670114323254093969066948162, −19.47445561166108617312252778061, −17.8081913358337836941162318015, −16.16109132006472214496456023856, −15.21856002411232212364290254571, −14.18819229928491524768179498881, −12.68465357576520696645030519707, −11.62946079532379366571376415106, −10.58884701132589883242144881856, −8.172052226013165936649236267298, −7.157198933695523433032469267166, −5.22163380154973557603048846383, −4.1837960607196309930620670359, −2.40313421671922817586739789725, 2.40313421671922817586739789725, 4.1837960607196309930620670359, 5.22163380154973557603048846383, 7.157198933695523433032469267166, 8.172052226013165936649236267298, 10.58884701132589883242144881856, 11.62946079532379366571376415106, 12.68465357576520696645030519707, 14.18819229928491524768179498881, 15.21856002411232212364290254571, 16.16109132006472214496456023856, 17.8081913358337836941162318015, 19.47445561166108617312252778061, 20.43670114323254093969066948162, 21.57822376673138383783744370346, 22.76715949979014002691448233615, 23.96150603262041493797356734403, 24.40192557124825977782532324401, 26.138076418826237379980199526152, 27.3242642506575048000894215764, 28.62351137889036648079428283193, 29.8578836602481893856332621913, 31.128217180029262240231189064165, 31.40445266752256121352949573385, 32.8182887332261932697096796825

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