Properties

Degree 1
Conductor 83
Sign $-0.991 + 0.131i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.477 + 0.878i)2-s + (−0.264 + 0.964i)3-s + (−0.543 + 0.839i)4-s + (−0.771 − 0.636i)5-s + (−0.973 + 0.227i)6-s + (−0.665 + 0.746i)7-s + (−0.997 − 0.0765i)8-s + (−0.859 − 0.511i)9-s + (0.190 − 0.981i)10-s + (0.817 + 0.575i)11-s + (−0.665 − 0.746i)12-s + (0.606 + 0.795i)13-s + (−0.973 − 0.227i)14-s + (0.817 − 0.575i)15-s + (−0.409 − 0.912i)16-s + (−0.927 − 0.373i)17-s + ⋯
L(s,χ)  = 1  + (0.477 + 0.878i)2-s + (−0.264 + 0.964i)3-s + (−0.543 + 0.839i)4-s + (−0.771 − 0.636i)5-s + (−0.973 + 0.227i)6-s + (−0.665 + 0.746i)7-s + (−0.997 − 0.0765i)8-s + (−0.859 − 0.511i)9-s + (0.190 − 0.981i)10-s + (0.817 + 0.575i)11-s + (−0.665 − 0.746i)12-s + (0.606 + 0.795i)13-s + (−0.973 − 0.227i)14-s + (0.817 − 0.575i)15-s + (−0.409 − 0.912i)16-s + (−0.927 − 0.373i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.991 + 0.131i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.991 + 0.131i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $-0.991 + 0.131i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (33, \cdot )$
Sato-Tate  :  $\mu(41)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 83,\ (0:\ ),\ -0.991 + 0.131i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.05218065242 + 0.7905776382i$
$L(\frac12,\chi)$  $\approx$  $0.05218065242 + 0.7905776382i$
$L(\chi,1)$  $\approx$  0.5496656295 + 0.7235095732i
$L(1,\chi)$  $\approx$  0.5496656295 + 0.7235095732i

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

−30.40514451611457770568614286608, −29.53729545314569268195402922331, −28.61847969144331011893273800864, −27.38264193177434622904122654138, −26.317403499563308402102476685717, −24.66257547117342909971811242043, −23.65105681274879752057046251699, −22.725176695075658719030643369552, −22.22596310178353875622516273957, −20.20406999105459973221980505443, −19.62099945567106554926675163251, −18.70000994333034342166705577667, −17.63687227050735378908484626358, −15.974058909552629249071988880803, −14.4057382596461191680007147496, −13.48368430785477614493983758725, −12.434130208426622933102168016317, −11.309599950215721890344321137, −10.52340557319031042422876407128, −8.64711427042757425744173241379, −7.03517808962717517281606840274, −5.98576381108846083005965430241, −3.98833868750031452099225337312, −2.84246514862309621577873834592, −0.815236042323645389927492298540, 3.457954087571255053816876163914, 4.453525170522151707296270738730, 5.64267948340131258768671896026, 7.00391253568421978409093548518, 8.817547161371300351088891684422, 9.33042083177724841010798311817, 11.55782268038262499611970983485, 12.315893311403804686756918526690, 13.88459024725852851862490584727, 15.284468281628507499815297652017, 15.88109497274240867055448296638, 16.66806721130211144936694465684, 17.94524481925143987750908429803, 19.64420415891106456185444380196, 20.87154423013596017536839552712, 22.053013793509794789071508344479, 22.75149705034792571308536237128, 23.807131680817353843286211799890, 25.018432298325488997353779232, 26.00488102913491256066678660633, 27.12547700974422107958813404674, 27.94404088736116588195230033414, 29.007860524926009247717788310744, 31.171689524395863445924802047708, 31.31627910457141149261718208336

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