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Properties

Label 1-179-179.12-r0-0-0
Degree $1$
Conductor $179$
Sign $0.0238 + 0.999i$
Analytic cond. $0.831272$
Root an. cond. $0.831272$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $0.0238 + 0.999i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ 0.0238 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1020398399 + 0.1045080772i\)
\(L(\frac12)\) \(\approx\) \(0.1020398399 + 0.1045080772i\)
\(L(1)\) \(\approx\) \(0.4402468103 - 0.1453014100i\)
\(L(1)\) \(\approx\) \(0.4402468103 - 0.1453014100i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (-0.737 - 0.675i)T \)
3 \( 1 + (-0.0529 - 0.998i)T \)
5 \( 1 + (-0.192 + 0.981i)T \)
7 \( 1 + (-0.458 - 0.888i)T \)
11 \( 1 + (-0.123 + 0.992i)T \)
13 \( 1 + (-0.949 + 0.312i)T \)
17 \( 1 + (-0.863 - 0.505i)T \)
19 \( 1 + (-0.688 + 0.725i)T \)
23 \( 1 + (-0.896 + 0.442i)T \)
29 \( 1 + (0.880 - 0.474i)T \)
31 \( 1 + (-0.394 + 0.918i)T \)
37 \( 1 + (0.880 + 0.474i)T \)
41 \( 1 + (0.227 - 0.973i)T \)
43 \( 1 + (-0.925 + 0.378i)T \)
47 \( 1 + (0.0176 + 0.999i)T \)
53 \( 1 + (0.158 - 0.987i)T \)
59 \( 1 + (0.713 - 0.700i)T \)
61 \( 1 + (-0.999 + 0.0352i)T \)
67 \( 1 + (0.607 + 0.794i)T \)
71 \( 1 + (-0.969 + 0.244i)T \)
73 \( 1 + (-0.783 + 0.621i)T \)
79 \( 1 + (0.295 - 0.955i)T \)
83 \( 1 + (0.489 - 0.871i)T \)
89 \( 1 + (-0.737 + 0.675i)T \)
97 \( 1 + (-0.825 + 0.564i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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