Properties

Degree 1
Conductor 83
Sign $0.323 + 0.946i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.190 − 0.981i)2-s + (0.338 − 0.941i)3-s + (−0.927 + 0.373i)4-s + (−0.896 + 0.443i)5-s + (−0.988 − 0.152i)6-s + (0.0383 − 0.999i)7-s + (0.543 + 0.839i)8-s + (−0.771 − 0.636i)9-s + (0.606 + 0.795i)10-s + (−0.114 + 0.993i)11-s + (0.0383 + 0.999i)12-s + (−0.817 + 0.575i)13-s + (−0.988 + 0.152i)14-s + (0.114 + 0.993i)15-s + (0.720 − 0.693i)16-s + (−0.264 + 0.964i)17-s + ⋯
L(s,χ)  = 1  + (−0.190 − 0.981i)2-s + (0.338 − 0.941i)3-s + (−0.927 + 0.373i)4-s + (−0.896 + 0.443i)5-s + (−0.988 − 0.152i)6-s + (0.0383 − 0.999i)7-s + (0.543 + 0.839i)8-s + (−0.771 − 0.636i)9-s + (0.606 + 0.795i)10-s + (−0.114 + 0.993i)11-s + (0.0383 + 0.999i)12-s + (−0.817 + 0.575i)13-s + (−0.988 + 0.152i)14-s + (0.114 + 0.993i)15-s + (0.720 − 0.693i)16-s + (−0.264 + 0.964i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.323 + 0.946i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.323 + 0.946i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $0.323 + 0.946i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (53, \cdot )$
Sato-Tate  :  $\mu(82)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 83,\ (1:\ ),\ 0.323 + 0.946i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.03634360428 + 0.02597644584i$
$L(\frac12,\chi)$  $\approx$  $0.03634360428 + 0.02597644584i$
$L(\chi,1)$  $\approx$  0.4931079220 - 0.3825728961i
$L(1,\chi)$  $\approx$  0.4931079220 - 0.3825728961i

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

−31.18590634802787455793310558730, −28.95535599920062084731124225733, −27.69930579039576542203047961980, −27.24375639098639353206802994390, −26.318532481053968386982002418112, −24.9163040843161115814678408989, −24.39492039940722105859716298581, −22.800072073124136278300782452307, −22.09590244289764406175123144990, −20.69154184357421645143273184392, −19.41023713001642311711342246107, −18.415578223602339437654565359095, −16.725626192025400990662278642502, −16.05016606222310892959185039893, −15.16818996852811801968896062876, −14.26886663323109589160616319982, −12.626774425175497353663313993217, −11.09362625856823330957556119197, −9.49582516177692915645970961488, −8.62761250238045651514934541372, −7.67433335861818951122313009343, −5.65960944383166677756489855862, −4.756029425805893103391751789956, −3.23161072523723009026952066463, −0.02171542601507234150847120667, 1.70219832464217839894271249808, 3.22325659006092448169596327381, 4.50737205597567693373492875341, 7.07225723442242884262223016985, 7.75428169021068877663700805889, 9.27433457605766934975185607496, 10.728478357444404921554554758773, 11.80237323189217386574649928801, 12.78719723180784362549030645096, 13.90502112745341403661108667552, 15.02915799209978187496400806781, 17.11754870523789957792518666185, 17.92059520726760648340700502240, 19.36109169358742236587290825196, 19.62618040737526435458887132804, 20.71447444142652812671317894305, 22.3370094107207067427848428316, 23.32333527374609555379324656108, 24.05065585625608857202221449341, 25.96981520069900437150050750345, 26.51618563580615449461385875957, 27.75983259666215784239272123969, 28.95197586000417993266403536595, 29.89151345618979349083415753621, 30.78173347844767507742445598490

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