Properties

Degree 1
Conductor 83
Sign $0.996 - 0.0870i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.997 − 0.0765i)2-s + (0.720 + 0.693i)3-s + (0.988 − 0.152i)4-s + (−0.477 − 0.878i)5-s + (0.771 + 0.636i)6-s + (0.817 − 0.575i)7-s + (0.973 − 0.227i)8-s + (0.0383 + 0.999i)9-s + (−0.543 − 0.839i)10-s + (−0.264 − 0.964i)11-s + (0.817 + 0.575i)12-s + (0.927 + 0.373i)13-s + (0.771 − 0.636i)14-s + (0.264 − 0.964i)15-s + (0.953 − 0.301i)16-s + (−0.409 + 0.912i)17-s + ⋯
L(s,χ)  = 1  + (0.997 − 0.0765i)2-s + (0.720 + 0.693i)3-s + (0.988 − 0.152i)4-s + (−0.477 − 0.878i)5-s + (0.771 + 0.636i)6-s + (0.817 − 0.575i)7-s + (0.973 − 0.227i)8-s + (0.0383 + 0.999i)9-s + (−0.543 − 0.839i)10-s + (−0.264 − 0.964i)11-s + (0.817 + 0.575i)12-s + (0.927 + 0.373i)13-s + (0.771 − 0.636i)14-s + (0.264 − 0.964i)15-s + (0.953 − 0.301i)16-s + (−0.409 + 0.912i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.996 - 0.0870i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.996 - 0.0870i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $0.996 - 0.0870i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (42, \cdot )$
Sato-Tate  :  $\mu(82)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 83,\ (1:\ ),\ 0.996 - 0.0870i)$
$L(\chi,\frac{1}{2})$  $\approx$  $3.840202456 - 0.1675559692i$
$L(\frac12,\chi)$  $\approx$  $3.840202456 - 0.1675559692i$
$L(\chi,1)$  $\approx$  2.402596793 + 0.03088052139i
$L(1,\chi)$  $\approx$  2.402596793 + 0.03088052139i

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

−30.79712527215908178644607836951, −30.16369500514851202620250065483, −28.872782544128616011636326041917, −27.37981540209007490352593717403, −25.8486255989362975813838825300, −25.31490053747021438412356273708, −24.04659931274902019529915819557, −23.283071016624047691156336378718, −22.17618887558518574199992888643, −20.82106673049098023916676710375, −20.07444165787092707513977073081, −18.68549529336190325222038895110, −17.82035264933825204981406178457, −15.68034408714217751321317805728, −14.93401528739742776513240778298, −14.12303055799551249629385370049, −12.86106802407405760645092693546, −11.82364748488900955458342145059, −10.68351595133405136921964936524, −8.51659767376576744628993694637, −7.37810565992567042494377555411, −6.404388873438888044030701776253, −4.61164435046516979853707518571, −3.07490662899518749421406074060, −2.00487831976325214828280790087, 1.686077153872408123365798856389, 3.6817176786792493706907652804, 4.35971598322255371828364683647, 5.726124835233169649076415367891, 7.77903007110180909047682281087, 8.685301247865897121186546402346, 10.590636301812591390706306101036, 11.43553643588415785132978022880, 13.09297847549956359376559948403, 13.8802912389769602428982584627, 15.06101084757489404834863846984, 16.03497496854141299589038487093, 16.92583815253885418505608103120, 19.21382607215245012665361363030, 20.105663735101229876829592142859, 21.1133967163119807148149512470, 21.52948747188468882126295275813, 23.32653888634109556476216247891, 24.00299347656526887290569324708, 25.063128234169449363439336000814, 26.28711838992642265472552249205, 27.47824065840633142711065359065, 28.43906074396113631786099436638, 29.88661466610117692279259134974, 30.960832068645450248619825885

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