Properties

Degree 1
Conductor 83
Sign $-0.674 + 0.738i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.859 + 0.511i)2-s + (0.606 + 0.795i)3-s + (0.477 + 0.878i)4-s + (−0.338 + 0.941i)5-s + (0.114 + 0.993i)6-s + (−0.409 − 0.912i)7-s + (−0.0383 + 0.999i)8-s + (−0.264 + 0.964i)9-s + (−0.771 + 0.636i)10-s + (0.953 + 0.301i)11-s + (−0.409 + 0.912i)12-s + (−0.896 − 0.443i)13-s + (0.114 − 0.993i)14-s + (−0.953 + 0.301i)15-s + (−0.543 + 0.839i)16-s + (0.190 − 0.981i)17-s + ⋯
L(s,χ)  = 1  + (0.859 + 0.511i)2-s + (0.606 + 0.795i)3-s + (0.477 + 0.878i)4-s + (−0.338 + 0.941i)5-s + (0.114 + 0.993i)6-s + (−0.409 − 0.912i)7-s + (−0.0383 + 0.999i)8-s + (−0.264 + 0.964i)9-s + (−0.771 + 0.636i)10-s + (0.953 + 0.301i)11-s + (−0.409 + 0.912i)12-s + (−0.896 − 0.443i)13-s + (0.114 − 0.993i)14-s + (−0.953 + 0.301i)15-s + (−0.543 + 0.839i)16-s + (0.190 − 0.981i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.674 + 0.738i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.674 + 0.738i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $-0.674 + 0.738i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (45, \cdot )$
Sato-Tate  :  $\mu(82)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 83,\ (1:\ ),\ -0.674 + 0.738i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.190627826 + 2.701703913i$
$L(\frac12,\chi)$  $\approx$  $1.190627826 + 2.701703913i$
$L(\chi,1)$  $\approx$  1.422416022 + 1.303980132i
$L(1,\chi)$  $\approx$  1.422416022 + 1.303980132i

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

−30.51594601873476134229251194483, −29.15111472844542517647512607292, −28.6118893909640142219867038056, −27.26349598483634921371422392946, −25.49381351695260610797132280096, −24.4800709256414030995755736193, −24.13798370883986925107063924592, −22.672208310863933486087019489866, −21.561973905244997024535445844267, −20.385881570901531773936924510481, −19.46034135929010792445497243088, −18.85467948607444152381093197844, −16.98956456413325461120846594776, −15.51061749759665674027097319618, −14.50067333954766343517305377531, −13.3228035070395106831249532899, −12.2501163180642180049638414412, −11.83415472830508323761921570241, −9.60530380827554693316163153644, −8.59449482649473331382325034444, −6.87025375928984258137322535572, −5.58647515334328137765028673573, −3.98871909635283908697027044948, −2.5507050098513907504744005248, −1.086249520778363380960601608811, 2.88637607302770311501198417840, 3.7294935965346957615149895356, 5.018160580152560026489749472913, 6.89771552828949544768061435618, 7.646258795327857433042390277104, 9.4876623286758008185243253067, 10.74754788991229773617634721761, 12.03773099841398710120824040013, 13.78692744594200478091995588376, 14.35220032059905132108601847584, 15.40298218164980078413829180246, 16.36271396364798942806310240498, 17.55331277468028003191039937488, 19.55950199579449735898795521509, 20.21656128450047312283319429860, 21.60967499889430383179827639473, 22.54441109065849329102358007232, 23.11124448633238648341257765705, 24.811607489912183894553152189803, 25.60240340558927067260969702690, 26.832704746768379788198915957693, 27.11315156957506612086512735673, 29.32251264491744650642689168389, 30.26451939226093076555833775243, 31.12850142545491899925853337208

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