Properties

Degree 1
Conductor 179
Sign $-0.999 + 0.00430i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.227 − 0.973i)2-s + (0.938 − 0.345i)3-s + (−0.896 − 0.442i)4-s + (−0.969 − 0.244i)5-s + (−0.123 − 0.992i)6-s + (−0.999 + 0.0352i)7-s + (−0.635 + 0.772i)8-s + (0.760 − 0.648i)9-s + (−0.458 + 0.888i)10-s + (0.295 − 0.955i)11-s + (−0.994 − 0.105i)12-s + (−0.520 − 0.854i)13-s + (−0.192 + 0.981i)14-s + (−0.994 + 0.105i)15-s + (0.607 + 0.794i)16-s + (−0.925 − 0.378i)17-s + ⋯
L(s,χ)  = 1  + (0.227 − 0.973i)2-s + (0.938 − 0.345i)3-s + (−0.896 − 0.442i)4-s + (−0.969 − 0.244i)5-s + (−0.123 − 0.992i)6-s + (−0.999 + 0.0352i)7-s + (−0.635 + 0.772i)8-s + (0.760 − 0.648i)9-s + (−0.458 + 0.888i)10-s + (0.295 − 0.955i)11-s + (−0.994 − 0.105i)12-s + (−0.520 − 0.854i)13-s + (−0.192 + 0.981i)14-s + (−0.994 + 0.105i)15-s + (0.607 + 0.794i)16-s + (−0.925 − 0.378i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.999 + 0.00430i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.999 + 0.00430i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(179\)
\( \varepsilon \)  =  $-0.999 + 0.00430i$
motivic weight  =  \(0\)
character  :  $\chi_{179} (20, \cdot )$
Sato-Tate  :  $\mu(89)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 179,\ (0:\ ),\ -0.999 + 0.00430i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.002067423459 - 0.9596606951i$
$L(\frac12,\chi)$  $\approx$  $0.002067423459 - 0.9596606951i$
$L(\chi,1)$  $\approx$  0.6482141873 - 0.7747441423i
$L(1,\chi)$  $\approx$  0.6482141873 - 0.7747441423i

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

−27.40366957912696832998072629355, −26.53722327406780821118918793642, −26.010909180718159790908076148815, −25.0906340538489093998838240244, −24.10040053348170125737750676247, −23.071741021417619196985608873341, −22.28469820663505722769596763898, −21.29414011961407769819566556015, −19.713221939486351302503471331860, −19.38478025505961511152865148195, −18.10564309981624075729040691297, −16.62467967042160921826269352209, −15.94605690113926659604136304831, −14.96502217363108873772458740713, −14.47387432527125174899354344439, −13.06047618107717739412871561762, −12.355354499824220617027938782585, −10.495454238022885084588509128391, −9.25024081586381759274943984301, −8.54992356023268629288848007026, −7.17292819568440276471137748310, −6.70147828324924841874198065695, −4.565837030146155574136795200279, −4.01096067732265672229626806135, −2.66903651568188858105398611207, 0.64230660054083535547663058103, 2.550385441132158698084597878037, 3.423582604554140006073487660803, 4.37328194401279800950607819218, 6.16566362271086937800093136925, 7.71058468507304328139343490320, 8.75165735079186926762533251614, 9.5799561501733421177966011371, 10.903111358006717151447606411313, 12.07107778206956977752584780704, 12.95319952049512319040594461084, 13.62778284022812109174414758074, 14.967034446501457013181751061068, 15.72056352770188819327397109084, 17.294259353698607108183079847682, 18.804165896218065015855672158202, 19.30509458270235485124818692352, 19.910777242798968987992570303207, 20.824492060037674350105569940517, 22.0129246823700652272212631170, 22.91833271405143810264082685523, 23.93598181280548625571653505830, 24.825766482476388066375126373079, 26.14419727788298787682934080362, 27.04709857649692337648197890368

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