Properties

Degree 1
Conductor 179
Sign $-0.525 + 0.851i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.688 + 0.725i)2-s + (0.960 + 0.278i)3-s + (−0.0529 − 0.998i)4-s + (0.489 + 0.871i)5-s + (−0.863 + 0.505i)6-s + (−0.825 + 0.564i)7-s + (0.760 + 0.648i)8-s + (0.844 + 0.535i)9-s + (−0.969 − 0.244i)10-s + (−0.925 − 0.378i)11-s + (0.227 − 0.973i)12-s + (−0.123 + 0.992i)13-s + (0.158 − 0.987i)14-s + (0.227 + 0.973i)15-s + (−0.994 + 0.105i)16-s + (−0.949 − 0.312i)17-s + ⋯
L(s,χ)  = 1  + (−0.688 + 0.725i)2-s + (0.960 + 0.278i)3-s + (−0.0529 − 0.998i)4-s + (0.489 + 0.871i)5-s + (−0.863 + 0.505i)6-s + (−0.825 + 0.564i)7-s + (0.760 + 0.648i)8-s + (0.844 + 0.535i)9-s + (−0.969 − 0.244i)10-s + (−0.925 − 0.378i)11-s + (0.227 − 0.973i)12-s + (−0.123 + 0.992i)13-s + (0.158 − 0.987i)14-s + (0.227 + 0.973i)15-s + (−0.994 + 0.105i)16-s + (−0.949 − 0.312i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.525 + 0.851i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.525 + 0.851i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(179\)
\( \varepsilon \)  =  $-0.525 + 0.851i$
motivic weight  =  \(0\)
character  :  $\chi_{179} (138, \cdot )$
Sato-Tate  :  $\mu(89)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 179,\ (0:\ ),\ -0.525 + 0.851i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.4994902485 + 0.8949873590i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.4994902485 + 0.8949873590i\)
\(L(\chi,1)\)  \(\approx\)  \(0.7793883938 + 0.5894812078i\)
\(L(1,\chi)\)  \(\approx\)  \(0.7793883938 + 0.5894812078i\)

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

−26.80074357762127867888920221093, −26.22926500759915704443451791149, −25.275240243865961657813590018817, −24.586071457977470656882310171127, −23.14524795672080715526246314557, −21.857937024192652051659575687751, −20.78691760402684500426200400221, −20.16133009879705709649536345416, −19.60702387390376137431898546190, −18.31353586242284326047969603328, −17.56675933229516038959741846282, −16.34961364810920192690589353977, −15.42236192024016351529447447624, −13.59748721206824027557166685152, −13.068500559970425199576772506044, −12.39227342914616424836248206400, −10.55392877885773234199666010573, −9.77863512586274147254529320576, −8.87785283340148690513998012258, −7.91981962784991733987958294205, −6.85789877687561530266903786043, −4.84231915709585926774994877006, −3.39851410252511034335628579025, −2.39593023337909153873160542962, −0.948997079844697178750254514522, 2.10146235787050060041016904328, 3.11069301689298949240111016339, 4.98265040473495067566589357860, 6.36697585024747277667794186720, 7.222489221676485379314344038365, 8.45940534832625873184308470341, 9.49577147929426140151874973880, 10.04895372184632329214003589179, 11.33209406691177749454822364221, 13.43391585518552997588647843715, 13.87860070170609058641422352789, 15.25260767326827399251789124385, 15.61781795372880229654825243093, 16.80200789415964471829253717870, 18.25524128278445064745009253173, 18.814312616986428608072332239653, 19.55037726687128810584318258608, 20.87404245091970411477028551499, 21.92203801920290912451322102999, 22.89190967428733258052912558792, 24.2807310395112427652510754287, 25.06906871682408026765697684395, 25.97580692103312309905630321099, 26.44838064475722550241916447043, 27.14337865233369003551323368397

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