Properties

Degree 1
Conductor $ 3 \cdot 19 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(57\)    =    \(3 \cdot 19\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{57} (56, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 57,\ (0:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.385011232$
$L(\frac12,\chi)$  $\approx$  $1.385011232$
$L(\chi,1)$  $\approx$  1.512726166
$L(1,\chi)$  $\approx$  1.512726166

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

−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