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  :  $\chi_{179} (12, \cdot )$
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

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

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