Properties

Degree 1
Conductor 179
Sign $0.691 - 0.722i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.896 − 0.442i)2-s + (0.760 − 0.648i)3-s + (0.607 + 0.794i)4-s + (0.880 + 0.474i)5-s + (−0.969 + 0.244i)6-s + (0.997 − 0.0705i)7-s + (−0.192 − 0.981i)8-s + (0.158 − 0.987i)9-s + (−0.579 − 0.815i)10-s + (−0.825 − 0.564i)11-s + (0.977 + 0.210i)12-s + (−0.458 + 0.888i)13-s + (−0.925 − 0.378i)14-s + (0.977 − 0.210i)15-s + (−0.261 + 0.965i)16-s + (0.713 + 0.700i)17-s + ⋯
L(s,χ)  = 1  + (−0.896 − 0.442i)2-s + (0.760 − 0.648i)3-s + (0.607 + 0.794i)4-s + (0.880 + 0.474i)5-s + (−0.969 + 0.244i)6-s + (0.997 − 0.0705i)7-s + (−0.192 − 0.981i)8-s + (0.158 − 0.987i)9-s + (−0.579 − 0.815i)10-s + (−0.825 − 0.564i)11-s + (0.977 + 0.210i)12-s + (−0.458 + 0.888i)13-s + (−0.925 − 0.378i)14-s + (0.977 − 0.210i)15-s + (−0.261 + 0.965i)16-s + (0.713 + 0.700i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.691 - 0.722i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.691 - 0.722i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(179\)
\( \varepsilon \)  =  $0.691 - 0.722i$
motivic weight  =  \(0\)
character  :  $\chi_{179} (42, \cdot )$
Sato-Tate  :  $\mu(89)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 179,\ (0:\ ),\ 0.691 - 0.722i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.093744070 - 0.4669257297i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.093744070 - 0.4669257297i\)
\(L(\chi,1)\)  \(\approx\)  \(1.030672374 - 0.3183153214i\)
\(L(1,\chi)\)  \(\approx\)  \(1.030672374 - 0.3183153214i\)

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

−27.49040611389773281197462493123, −26.436755355311138384479187777042, −25.640458645653745418520750980810, −24.812352552898436154203578370935, −24.23111866915958493945290147050, −22.65148429519605213478239042104, −21.24308768794348399078334004835, −20.55893684849314191110222771364, −20.01879963799232493643809557881, −18.37239425761518979415692139261, −17.874659063970580623450500964030, −16.650412285530046487964128190404, −15.84724893259343905676593214135, −14.6521823866135866126571050295, −14.14570039684929746166240727241, −12.62416603510980030952957333779, −10.93879645979767896786582986772, −10.01704643485393047180862705489, −9.33361397016863527622207837501, −8.133891677314020884398896847293, −7.51159399760337526863928639253, −5.48924468538617207128075977637, −4.95031128225596540161487365782, −2.72425685139412943866075221034, −1.621906973360408031990774412384, 1.52690122797189877824312324692, 2.30795671892030473201487620979, 3.56928551275167595844767104389, 5.69798380734783545811430205395, 7.16291782855990982541940105517, 7.873447246530207332427068267698, 9.00180067460165964112479468601, 9.932315423204318146171240998271, 11.08350541589362928037882415947, 12.13113538316665492047749511823, 13.43109553088658882501664931034, 14.19291884993467077945219601111, 15.36758119689256449972783202961, 16.899845685710059106668968915598, 17.78916220874823494782078712912, 18.52559482948048819008783361593, 19.21751115281928650761272023209, 20.49541067015644889701100479492, 21.15702478186030369522198203444, 21.92958742118330444294667870356, 23.91034629919342478819983981053, 24.459266288561746703689474589658, 25.657058351006564406306010265347, 26.223266151011410621619795075805, 26.96787385959491827654977242713

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