Properties

Degree 1
Conductor 179
Sign $0.0238 - 0.999i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.737 + 0.675i)2-s + (−0.0529 + 0.998i)3-s + (0.0881 − 0.996i)4-s + (−0.192 − 0.981i)5-s + (−0.635 − 0.772i)6-s + (−0.458 + 0.888i)7-s + (0.607 + 0.794i)8-s + (−0.994 − 0.105i)9-s + (0.804 + 0.593i)10-s + (−0.123 − 0.992i)11-s + (0.990 + 0.140i)12-s + (−0.949 − 0.312i)13-s + (−0.261 − 0.965i)14-s + (0.990 − 0.140i)15-s + (−0.984 − 0.175i)16-s + (−0.863 + 0.505i)17-s + ⋯
L(s,χ)  = 1  + (−0.737 + 0.675i)2-s + (−0.0529 + 0.998i)3-s + (0.0881 − 0.996i)4-s + (−0.192 − 0.981i)5-s + (−0.635 − 0.772i)6-s + (−0.458 + 0.888i)7-s + (0.607 + 0.794i)8-s + (−0.994 − 0.105i)9-s + (0.804 + 0.593i)10-s + (−0.123 − 0.992i)11-s + (0.990 + 0.140i)12-s + (−0.949 − 0.312i)13-s + (−0.261 − 0.965i)14-s + (0.990 − 0.140i)15-s + (−0.984 − 0.175i)16-s + (−0.863 + 0.505i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.0238 - 0.999i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.0238 - 0.999i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(179\)
\( \varepsilon \)  =  $0.0238 - 0.999i$
motivic weight  =  \(0\)
character  :  $\chi_{179} (15, \cdot )$
Sato-Tate  :  $\mu(89)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 179,\ (0:\ ),\ 0.0238 - 0.999i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.1020398399 - 0.1045080772i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.1020398399 - 0.1045080772i\)
\(L(\chi,1)\)  \(\approx\)  \(0.4402468103 + 0.1453014100i\)
\(L(1,\chi)\)  \(\approx\)  \(0.4402468103 + 0.1453014100i\)

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

−27.52270692383124594017201843152, −26.61877892411695121606231608151, −25.83896695788370177224024974510, −25.05729368841150475280326655505, −23.58407681006695515553564689875, −22.77716544744934955343188783250, −21.92711113160999835903452383598, −20.36755639576101166352134749511, −19.64962217048545578889563240350, −18.99606216049724954723029871499, −17.86101045486727955504366469101, −17.41352849277504879939053790080, −16.12023132563384699092460460660, −14.56302507879590474117888038908, −13.51612863362644805888387373594, −12.466557811961847948832651799152, −11.609918419682345846166165824711, −10.495121177074267127305101081995, −9.66354446781178394847238096363, −8.048993709887344011018159712433, −7.18680878851391764670655635041, −6.592110279231994149772923102498, −4.247478358529787206563888490928, −2.839879522287982065633625116594, −1.85664135082861217811186439409, 0.13863873452957946026196351059, 2.48733966466051495142332406590, 4.37274556427797915248008344069, 5.401307073725187253189238931758, 6.28422607688844179537855742799, 8.17232978236026200407608115892, 8.82184976196363281907703087583, 9.62453411876084931999945167730, 10.76472708955663737772788630783, 11.92234096225748640867980424546, 13.34062025592411042367979871039, 14.82170133611657121426841874404, 15.52517170086666953325448798987, 16.36203700073348423333646982171, 16.9987454139249084208402523173, 18.15365577151631740723487409612, 19.59069801041498347015112366824, 19.96553255350344498093901181257, 21.46298588972573188409983353530, 22.15160738931700235143010467290, 23.54620960277874704923194176631, 24.42969358750833332759452396596, 25.25114135792090621224128189761, 26.31980578444829109238061028328, 27.05803101638177455563781740839

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