Properties

Degree 1
Conductor 83
Sign $0.999 + 0.0357i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.973 + 0.227i)2-s + (−0.665 − 0.746i)3-s + (0.896 + 0.443i)4-s + (0.997 − 0.0765i)5-s + (−0.477 − 0.878i)6-s + (−0.264 + 0.964i)7-s + (0.771 + 0.636i)8-s + (−0.114 + 0.993i)9-s + (0.988 + 0.152i)10-s + (0.720 − 0.693i)11-s + (−0.264 − 0.964i)12-s + (0.409 − 0.912i)13-s + (−0.477 + 0.878i)14-s + (−0.720 − 0.693i)15-s + (0.606 + 0.795i)16-s + (0.953 + 0.301i)17-s + ⋯
L(s,χ)  = 1  + (0.973 + 0.227i)2-s + (−0.665 − 0.746i)3-s + (0.896 + 0.443i)4-s + (0.997 − 0.0765i)5-s + (−0.477 − 0.878i)6-s + (−0.264 + 0.964i)7-s + (0.771 + 0.636i)8-s + (−0.114 + 0.993i)9-s + (0.988 + 0.152i)10-s + (0.720 − 0.693i)11-s + (−0.264 − 0.964i)12-s + (0.409 − 0.912i)13-s + (−0.477 + 0.878i)14-s + (−0.720 − 0.693i)15-s + (0.606 + 0.795i)16-s + (0.953 + 0.301i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.999 + 0.0357i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.999 + 0.0357i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $0.999 + 0.0357i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (8, \cdot )$
Sato-Tate  :  $\mu(82)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 83,\ (1:\ ),\ 0.999 + 0.0357i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.973208408 + 0.05309153821i$
$L(\frac12,\chi)$  $\approx$  $2.973208408 + 0.05309153821i$
$L(\chi,1)$  $\approx$  1.908268340 + 0.01349561648i
$L(1,\chi)$  $\approx$  1.908268340 + 0.01349561648i

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

−30.44137513001086227897968314359, −29.51050891533872888724332743445, −28.81823519651232740691220940318, −27.696159198102890331501155738788, −26.22127726450686666664176689170, −25.25201725401018937804548373437, −23.81795636422831726447413311324, −22.84126905094692185547491075141, −22.164717300921636147903810316796, −20.92193792240810548529649227808, −20.468204607645597018604330715310, −18.6771470332179210489874124637, −16.89350384910828506669748597022, −16.57357698991859585345211391553, −14.83551799871569545390473936907, −14.05631644986247265729515977372, −12.750153550514575501975281755143, −11.52054948909007493465946988158, −10.32227964503330027277949753125, −9.597106789487603901914409226512, −6.89951893266405455327316303528, −6.003705834185768994640098208146, −4.65260072262297441153546965361, −3.57669649390367554429146514695, −1.49424387443532827278191121875, 1.594725537207360282347421405118, 3.10092291719965498103891906667, 5.44735333656085038171739444127, 5.81144207619431534441978763546, 7.08039922146219402407355655753, 8.755224777804115039000023941654, 10.680368009041089493370505361113, 11.87484291573287236374551295314, 12.89564206238601557066020022464, 13.66229207384481521316977719865, 14.994773451911610164045181388961, 16.405075682562330524283581062819, 17.33105210256142202632311840775, 18.50900679693267573965209574949, 19.847078796051645348384606669099, 21.52405724951079803273533472109, 21.9731629424758704929982566979, 23.0901572573830878432030534858, 24.24434695733207318916982230603, 25.126624535558243386321706483288, 25.64903705103990152126435623745, 27.84814664597657552919365434605, 28.82769493782076239641479577785, 29.81516877330753145695920991240, 30.33487224132279718413082917771

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