Properties

Degree 1
Conductor 179
Sign $0.877 + 0.478i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.949 − 0.312i)2-s + (−0.969 + 0.244i)3-s + (0.804 + 0.593i)4-s + (0.990 − 0.140i)5-s + (0.997 + 0.0705i)6-s + (0.607 + 0.794i)7-s + (−0.579 − 0.815i)8-s + (0.880 − 0.474i)9-s + (−0.984 − 0.175i)10-s + (−0.0529 + 0.998i)11-s + (−0.925 − 0.378i)12-s + (0.0881 − 0.996i)13-s + (−0.329 − 0.944i)14-s + (−0.925 + 0.378i)15-s + (0.295 + 0.955i)16-s + (−0.783 + 0.621i)17-s + ⋯
L(s,χ)  = 1  + (−0.949 − 0.312i)2-s + (−0.969 + 0.244i)3-s + (0.804 + 0.593i)4-s + (0.990 − 0.140i)5-s + (0.997 + 0.0705i)6-s + (0.607 + 0.794i)7-s + (−0.579 − 0.815i)8-s + (0.880 − 0.474i)9-s + (−0.984 − 0.175i)10-s + (−0.0529 + 0.998i)11-s + (−0.925 − 0.378i)12-s + (0.0881 − 0.996i)13-s + (−0.329 − 0.944i)14-s + (−0.925 + 0.378i)15-s + (0.295 + 0.955i)16-s + (−0.783 + 0.621i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.877 + 0.478i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.877 + 0.478i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(179\)
\( \varepsilon \)  =  $0.877 + 0.478i$
motivic weight  =  \(0\)
character  :  $\chi_{179} (25, \cdot )$
Sato-Tate  :  $\mu(89)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 179,\ (0:\ ),\ 0.877 + 0.478i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6684557457 + 0.1704226368i$
$L(\frac12,\chi)$  $\approx$  $0.6684557457 + 0.1704226368i$
$L(\chi,1)$  $\approx$  0.6767092122 + 0.05665887559i
$L(1,\chi)$  $\approx$  0.6767092122 + 0.05665887559i

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

−27.02474855046772731233495575923, −26.67568027516299156751473039705, −25.33319817185243729844324977970, −24.318812764778994211173363600559, −23.86666931577817543702466246506, −22.5523283886460402128971731089, −21.36698672508946916811988006972, −20.58670230561928472333201556841, −19.06477504893909501618727214593, −18.32060234334901329029524347535, −17.4892974688213318304255777542, −16.73260605058569327700722145558, −16.06485861017129081470856370295, −14.34366668271720350044662528901, −13.61156797863523637145808103857, −11.94324091676436695849326605395, −10.91977159924133035721256327055, −10.335822516129625285381158977, −9.093617907309365312866286698437, −7.78334274963052469432213801198, −6.59153111943868986343446743098, −5.96950100269299665702032611652, −4.59690614755730089342999629630, −2.18824498851023806412703964363, −0.9864559567603895938373107833, 1.37224815995267025318155020914, 2.55479254913837748842260334619, 4.65174298364390478075190235332, 5.81713212633248080393606437796, 6.830265207951098751919420557084, 8.29882259231721562719138967773, 9.470379144527219713348532233609, 10.26103554958450750020185773926, 11.21587542671858760547609382120, 12.26593385737231878034194530296, 13.11045287298420129038054552138, 15.106277251097566224354219394317, 15.757959617880067660994715316, 17.17870861583800999121861910166, 17.8405085566785106079672653368, 18.03880333890808705960214183986, 19.67002389893450457066871997952, 20.78447924798224403549211796580, 21.61665916880240092671859308123, 22.210692893836833735927527269757, 23.735287877782232238964230941332, 24.848243111801560458902097483166, 25.49225552883099097434044048516, 26.66405413395094709468736604300, 27.7635542387322234820570440262

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