Properties

Degree 1
Conductor 83
Sign $0.722 + 0.691i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.896 + 0.443i)2-s + (−0.114 − 0.993i)3-s + (0.606 − 0.795i)4-s + (−0.988 − 0.152i)5-s + (0.543 + 0.839i)6-s + (−0.859 + 0.511i)7-s + (−0.190 + 0.981i)8-s + (−0.973 + 0.227i)9-s + (0.953 − 0.301i)10-s + (0.0383 + 0.999i)11-s + (−0.859 − 0.511i)12-s + (0.665 − 0.746i)13-s + (0.543 − 0.839i)14-s + (−0.0383 + 0.999i)15-s + (−0.264 − 0.964i)16-s + (0.817 − 0.575i)17-s + ⋯
L(s,χ)  = 1  + (−0.896 + 0.443i)2-s + (−0.114 − 0.993i)3-s + (0.606 − 0.795i)4-s + (−0.988 − 0.152i)5-s + (0.543 + 0.839i)6-s + (−0.859 + 0.511i)7-s + (−0.190 + 0.981i)8-s + (−0.973 + 0.227i)9-s + (0.953 − 0.301i)10-s + (0.0383 + 0.999i)11-s + (−0.859 − 0.511i)12-s + (0.665 − 0.746i)13-s + (0.543 − 0.839i)14-s + (−0.0383 + 0.999i)15-s + (−0.264 − 0.964i)16-s + (0.817 − 0.575i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.722 + 0.691i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.722 + 0.691i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $0.722 + 0.691i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (35, \cdot )$
Sato-Tate  :  $\mu(82)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 83,\ (1:\ ),\ 0.722 + 0.691i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5526150438 + 0.2218802933i$
$L(\frac12,\chi)$  $\approx$  $0.5526150438 + 0.2218802933i$
$L(\chi,1)$  $\approx$  0.5473594829 + 0.005196438863i
$L(1,\chi)$  $\approx$  0.5473594829 + 0.005196438863i

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

−30.34487126218917199369687489667, −29.129708943422800669731152273300, −28.1915283719327566343909682983, −27.302514371026392314634481635232, −26.37197775751347650807017793651, −25.90306281851242465814407286054, −24.01100656305334524684264794122, −22.75440310868397826905463079807, −21.689461615644059750390691956276, −20.5988849902770365634295616584, −19.56281197270888616944462948201, −18.83508131574644573946933246870, −17.12903664124472572654528521193, −16.1565160662681439546124674006, −15.682520274711765385034232606318, −13.833117027541599706278341730426, −12.008579332395629514837327617210, −11.14892145712383652807266474048, −10.12176234423922969394819795182, −8.98755225461083778706740530818, −7.8024149444974703650046575571, −6.23192794031584317558005652745, −3.91874088540943991848974924961, −3.25681965088437815854486061183, −0.49760674411896227781960664951, 1.02706488827523875516481092223, 2.936602357181256854000209838007, 5.45892004953680217802852455032, 6.78704262754742530340083464472, 7.722765605882264711727764746195, 8.7627080396409065122006971388, 10.24687679943163858937946544043, 11.818092620545026244608987418628, 12.54529141737745796425198957514, 14.30855396172155546633284812234, 15.65781678825224891234969125110, 16.42490049855871965890134380489, 17.95397129870023779859295456221, 18.588535271526022087207164891420, 19.698638231982730891513673048943, 20.33704593264987757495642980038, 22.85023084279703751660038598654, 23.2369669607414338433611597927, 24.65886585762928989349962127139, 25.3042196545661473983333077230, 26.34180108751216554769476620273, 27.82487408153415603945308591200, 28.359825462868800230774687026512, 29.49449569313865577285061935337, 30.649939376185218009311366601904

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