Properties

Degree 1
Conductor $ 2^{4} \cdot 5 $
Sign $0.987 - 0.160i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + 3-s i·7-s + 9-s + i·11-s − 13-s i·17-s + i·19-s i·21-s + i·23-s + 27-s i·29-s − 31-s + i·33-s − 37-s − 39-s + ⋯
L(s,χ)  = 1  + 3-s i·7-s + 9-s + i·11-s − 13-s i·17-s + i·19-s i·21-s + i·23-s + 27-s i·29-s − 31-s + i·33-s − 37-s − 39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.987 - 0.160i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 80 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.987 - 0.160i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(80\)    =    \(2^{4} \cdot 5\)
\( \varepsilon \)  =  $0.987 - 0.160i$
motivic weight  =  \(0\)
character  :  $\chi_{80} (3, \cdot )$
Sato-Tate  :  $\mu(4)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 80,\ (0:\ ),\ 0.987 - 0.160i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.256569246 - 0.1012940218i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.256569246 - 0.1012940218i\)
\(L(\chi,1)\)  \(\approx\)  \(1.300878948 - 0.06371947111i\)
\(L(1,\chi)\)  \(\approx\)  \(1.300878948 - 0.06371947111i\)

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

−31.22211684378438150742387198463, −30.2121946371932238783921335298, −29.04888135712363536394778721722, −27.77254318457701260637761382705, −26.6912354980227638351639312292, −25.78260755812807073093095938132, −24.63875701269362329776613925180, −24.007900543114771189895370909705, −21.9845526225858333616046509213, −21.5373535950618278622856710911, −20.06282256025485464809963208367, −19.1722550781691026754372442464, −18.24224211947875785558586989120, −16.61817529622017202018017978584, −15.31503489860425835620765317178, −14.54597980318294571545935317393, −13.23533335827136056596492406287, −12.14272144358779081508855793667, −10.517107785521342553731410278748, −9.07086811199259908788522758982, −8.34462144976619229675821321962, −6.79411334932139178655355693864, −5.12493611766135929449494574875, −3.37915952862833272581648625561, −2.14188797151303214127298530970, 1.89529916722627262351051662045, 3.53070561316865122763017226892, 4.80988918647143802088166890276, 7.06949902679163590776709668165, 7.77279003615778496570699863342, 9.46407308624763364953057040016, 10.22122068429988564963900231625, 12.050038494582874104451291858470, 13.339234354818173448544041464309, 14.29968882987436340954179543175, 15.30639713007402057690491715424, 16.68179962022588176151403382052, 17.92865800278483745952376481558, 19.28822952619682017586149413603, 20.19605930642581084831330716711, 20.94100812527287843801662767478, 22.43052904290868607267417096772, 23.58860177470804919568349164453, 24.80600706963828876879014288813, 25.675865628711337013310212065480, 26.78822325481701119582064912752, 27.47722986896765903282357696084, 29.21587788564283151298322562228, 30.025176858272000245706580775770, 31.140493160461236874267680108334

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