Properties

Degree 1
Conductor $ 3 \cdot 19 $
Sign $-0.671 - 0.740i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s + 7-s − 8-s + (−0.5 − 0.866i)10-s − 11-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s − 20-s + (−0.5 + 0.866i)22-s + (0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s − 26-s + ⋯
L(s,χ)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s + 7-s − 8-s + (−0.5 − 0.866i)10-s − 11-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s − 20-s + (−0.5 + 0.866i)22-s + (0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s − 26-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.671 - 0.740i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.671 - 0.740i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(57\)    =    \(3 \cdot 19\)
\( \varepsilon \)  =  $-0.671 - 0.740i$
motivic weight  =  \(0\)
character  :  $\chi_{57} (26, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 57,\ (1:\ ),\ -0.671 - 0.740i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8119985845 - 1.832188669i$
$L(\frac12,\chi)$  $\approx$  $0.8119985845 - 1.832188669i$
$L(\chi,1)$  $\approx$  1.067915846 - 0.9681772550i
$L(1,\chi)$  $\approx$  1.067915846 - 0.9681772550i

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

−33.37459266749032514818082899074, −32.00303150340570409662392255355, −30.841569436562295669636982343183, −30.16154848132711754910970876969, −28.62560565435512977619231895651, −26.90137077488091019530496316096, −26.30543871366964565222571736461, −25.08166683644268962194945038140, −23.97260210265733261574785231630, −23.01326464575640717671396774324, −21.628779321650693107866161515676, −21.05634230838466790642720891109, −18.811500162662461169481304242461, −17.81850021202068781592838928715, −16.77751512088941891006265840560, −15.168212602889338865820207336793, −14.42577606456676618430191017742, −13.31584862648504745166393665994, −11.72991160709419412807743232294, −10.199401347217629675152279606701, −8.39054945198850491026151301607, −7.20136394785676526263100763442, −5.84944553066873169228082911939, −4.4759917534321526622685442712, −2.5488432379526665452185424833, 1.02702521794865644540331165104, 2.67098374995128088560317158195, 4.77321373331126485023358308620, 5.48545729013457190449960040377, 7.982326679377617106665930952676, 9.47245055206400385515295207352, 10.69710220749105265238668222275, 12.06575007244866482812135002651, 13.135214330097599173786759724, 14.22140615370001944579598641439, 15.610684052661416659955322483643, 17.41383123532230284238353038478, 18.35498629680616734541451677802, 19.972762737774123698712663276761, 20.859563669931778789989846291952, 21.573760275458204909418201753449, 23.11330234908604359073834969944, 24.12520899658294230668393126305, 25.1460887972799216650389048666, 27.082588301456514617246007932331, 27.92347278730336446017000157651, 29.03947359003603324621965388867, 29.89275706866518263573340082269, 31.22395769943453396678035535278, 31.97875476826675965947481656666

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