Properties

Degree 1
Conductor 179
Sign $-0.356 - 0.934i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.844 − 0.535i)2-s + (−0.261 + 0.965i)3-s + (0.427 − 0.904i)4-s + (−0.825 − 0.564i)5-s + (0.295 + 0.955i)6-s + (−0.688 − 0.725i)7-s + (−0.123 − 0.992i)8-s + (−0.863 − 0.505i)9-s + (−0.999 − 0.0352i)10-s + (−0.579 − 0.815i)11-s + (0.760 + 0.648i)12-s + (0.0176 − 0.999i)13-s + (−0.969 − 0.244i)14-s + (0.760 − 0.648i)15-s + (−0.635 − 0.772i)16-s + (0.880 + 0.474i)17-s + ⋯
L(s,χ)  = 1  + (0.844 − 0.535i)2-s + (−0.261 + 0.965i)3-s + (0.427 − 0.904i)4-s + (−0.825 − 0.564i)5-s + (0.295 + 0.955i)6-s + (−0.688 − 0.725i)7-s + (−0.123 − 0.992i)8-s + (−0.863 − 0.505i)9-s + (−0.999 − 0.0352i)10-s + (−0.579 − 0.815i)11-s + (0.760 + 0.648i)12-s + (0.0176 − 0.999i)13-s + (−0.969 − 0.244i)14-s + (0.760 − 0.648i)15-s + (−0.635 − 0.772i)16-s + (0.880 + 0.474i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.356 - 0.934i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.356 - 0.934i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(179\)
\( \varepsilon \)  =  $-0.356 - 0.934i$
motivic weight  =  \(0\)
character  :  $\chi_{179} (57, \cdot )$
Sato-Tate  :  $\mu(89)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 179,\ (0:\ ),\ -0.356 - 0.934i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6301571598 - 0.9144647577i$
$L(\frac12,\chi)$  $\approx$  $0.6301571598 - 0.9144647577i$
$L(\chi,1)$  $\approx$  1.021664299 - 0.4956146408i
$L(1,\chi)$  $\approx$  1.021664299 - 0.4956146408i

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

−27.73158965941525116005213572602, −26.049025481263645766463926553785, −25.79071238286097608440622058537, −24.52331816599414827089614721502, −23.75955508686439599717915849311, −22.87648262929664414397699273192, −22.42479193302977345332947058411, −21.109487372397832111506454253678, −19.77594241993248812125883617677, −18.8537975463515193894473745846, −17.98758011912849032434138512561, −16.68533364260704800992305152790, −15.735394159541298669673686131668, −14.85384324862704145294622858110, −13.741833416365919632153138555017, −12.72392169795805902340325017740, −11.93845885326930777300877387268, −11.19982635891943490399299627801, −9.20304516154919106858231153926, −7.64771314699293426128422815529, −7.18176169492355086851824278458, −6.090403889321776024642508229948, −4.941990337329764676345027753585, −3.33332227917359695142826941515, −2.3226601132837582966805883557, 0.69347620064815732064262727668, 3.30243635026352891253549503587, 3.67308657553394785707839874862, 5.06333533320852034038827462038, 5.85217663446623126054670313020, 7.55787611525281197416756883449, 9.04987095224155199403050168102, 10.40347845813840466732045202947, 10.83154994116126575269811432798, 12.18324576342136206295348573133, 12.94303811598797241979198343383, 14.23596809518933378171623680512, 15.27716388849484737712531086977, 16.18365517545702950729733894925, 16.72880668685882648050959508742, 18.636700787569710450620941495313, 19.75921325677041271242469606938, 20.44865638998100588679731345542, 21.17779620887638941695735342934, 22.38827045747025921882063498907, 23.06961310498964503764150944278, 23.72167086545080178566989866917, 24.95003499169824556924941731365, 26.336050199435979778692601895485, 27.290414482294749848602326714220

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