Properties

Degree 1
Conductor 83
Sign $-0.998 - 0.0559i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.409 − 0.912i)2-s + (0.477 − 0.878i)3-s + (−0.665 − 0.746i)4-s + (0.927 + 0.373i)5-s + (−0.606 − 0.795i)6-s + (−0.973 − 0.227i)7-s + (−0.953 + 0.301i)8-s + (−0.543 − 0.839i)9-s + (0.720 − 0.693i)10-s + (−0.771 − 0.636i)11-s + (−0.973 + 0.227i)12-s + (0.859 − 0.511i)13-s + (−0.606 + 0.795i)14-s + (0.771 − 0.636i)15-s + (−0.114 + 0.993i)16-s + (0.0383 − 0.999i)17-s + ⋯
L(s,χ)  = 1  + (0.409 − 0.912i)2-s + (0.477 − 0.878i)3-s + (−0.665 − 0.746i)4-s + (0.927 + 0.373i)5-s + (−0.606 − 0.795i)6-s + (−0.973 − 0.227i)7-s + (−0.953 + 0.301i)8-s + (−0.543 − 0.839i)9-s + (0.720 − 0.693i)10-s + (−0.771 − 0.636i)11-s + (−0.973 + 0.227i)12-s + (0.859 − 0.511i)13-s + (−0.606 + 0.795i)14-s + (0.771 − 0.636i)15-s + (−0.114 + 0.993i)16-s + (0.0383 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.998 - 0.0559i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.998 - 0.0559i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $-0.998 - 0.0559i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (39, \cdot )$
Sato-Tate  :  $\mu(82)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 83,\ (1:\ ),\ -0.998 - 0.0559i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.05534961816 - 1.978068603i$
$L(\frac12,\chi)$  $\approx$  $0.05534961816 - 1.978068603i$
$L(\chi,1)$  $\approx$  0.8335544702 - 1.124392697i
$L(1,\chi)$  $\approx$  0.8335544702 - 1.124392697i

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

−31.540132409427233164383770033791, −30.46323322113114835158579021762, −28.76984618968998950740326312915, −27.92972065171304962254312129046, −26.2789510936671146896434294913, −25.84018675971985109077188531429, −25.12093325343534309613331120764, −23.62908203625135296001759467113, −22.51095080146738706180984181456, −21.46456839964917737667443590803, −20.85047968253576771935047853979, −19.17134639903005649327591992249, −17.65464739401074570577231809090, −16.58060162611383119389745504585, −15.73294991384788805048273892675, −14.72395039660374664024085885028, −13.43979564936025012013497816933, −12.76190833459240468621010703962, −10.470679870511257727603867447207, −9.28581688729608848642223277899, −8.4941685074694431556647519254, −6.62988412532143723787404042456, −5.461433021809974990828846124157, −4.22999328967832849822596374545, −2.7126206967791472244662059017, 0.78164668315724642541374769581, 2.480061201795149186082850792951, 3.34774284299989966411637770482, 5.62693533962236274410605564602, 6.63409275233973510434089763882, 8.551881123571213247418719076486, 9.79431388455814037988312572804, 10.88812489470269764264956668427, 12.4664329474299702003552983607, 13.47464229853905200701385744848, 13.82867108794741535950745834423, 15.41521120931030010123993238155, 17.358086052584285287713802890561, 18.618087573147046980021970541170, 19.02542334357243733245265420745, 20.48092402427894381396174734025, 21.21001606000930999022471596267, 22.69918550458880134832127635952, 23.32794084466718228538582783228, 24.803238347991754995412654085921, 25.75337658626745624291849812500, 26.83309665259768272991527588386, 28.57343477015451465632300013077, 29.38184775162455223281911865596, 29.83050413143680804447719451944

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