Properties

Label 1-177-177.68-r1-0-0
Degree $1$
Conductor $177$
Sign $0.175 - 0.984i$
Analytic cond. $19.0212$
Root an. cond. $19.0212$
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  + (0.161 + 0.986i)2-s + (−0.947 + 0.319i)4-s + (0.561 − 0.827i)5-s + (0.976 + 0.214i)7-s + (−0.468 − 0.883i)8-s + (0.907 + 0.419i)10-s + (−0.796 − 0.605i)11-s + (−0.856 − 0.515i)13-s + (−0.0541 + 0.998i)14-s + (0.796 − 0.605i)16-s + (−0.976 + 0.214i)17-s + (−0.994 − 0.108i)19-s + (−0.267 + 0.963i)20-s + (0.468 − 0.883i)22-s + (−0.647 + 0.762i)23-s + ⋯
L(s)  = 1  + (0.161 + 0.986i)2-s + (−0.947 + 0.319i)4-s + (0.561 − 0.827i)5-s + (0.976 + 0.214i)7-s + (−0.468 − 0.883i)8-s + (0.907 + 0.419i)10-s + (−0.796 − 0.605i)11-s + (−0.856 − 0.515i)13-s + (−0.0541 + 0.998i)14-s + (0.796 − 0.605i)16-s + (−0.976 + 0.214i)17-s + (−0.994 − 0.108i)19-s + (−0.267 + 0.963i)20-s + (0.468 − 0.883i)22-s + (−0.647 + 0.762i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.175 - 0.984i$
Analytic conductor: \(19.0212\)
Root analytic conductor: \(19.0212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (1:\ ),\ 0.175 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6693714892 - 0.5605425900i\)
\(L(\frac12)\) \(\approx\) \(0.6693714892 - 0.5605425900i\)
\(L(1)\) \(\approx\) \(0.9189528106 + 0.1707354010i\)
\(L(1)\) \(\approx\) \(0.9189528106 + 0.1707354010i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 \)
good2 \( 1 + (0.161 + 0.986i)T \)
5 \( 1 + (0.561 - 0.827i)T \)
7 \( 1 + (0.976 + 0.214i)T \)
11 \( 1 + (-0.796 - 0.605i)T \)
13 \( 1 + (-0.856 - 0.515i)T \)
17 \( 1 + (-0.976 + 0.214i)T \)
19 \( 1 + (-0.994 - 0.108i)T \)
23 \( 1 + (-0.647 + 0.762i)T \)
29 \( 1 + (0.161 - 0.986i)T \)
31 \( 1 + (-0.994 + 0.108i)T \)
37 \( 1 + (0.468 - 0.883i)T \)
41 \( 1 + (-0.647 - 0.762i)T \)
43 \( 1 + (0.796 - 0.605i)T \)
47 \( 1 + (0.561 + 0.827i)T \)
53 \( 1 + (-0.907 + 0.419i)T \)
61 \( 1 + (-0.161 - 0.986i)T \)
67 \( 1 + (0.468 + 0.883i)T \)
71 \( 1 + (0.561 + 0.827i)T \)
73 \( 1 + (0.0541 - 0.998i)T \)
79 \( 1 + (0.267 - 0.963i)T \)
83 \( 1 + (0.725 - 0.687i)T \)
89 \( 1 + (0.161 - 0.986i)T \)
97 \( 1 + (0.0541 + 0.998i)T \)
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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

−27.35991171785061693612411617624, −26.651017199755203135725513611573, −25.70699758533904182069920427141, −24.211175527991669411624617092877, −23.40943442811298195656742905793, −22.23950037808921082723826177033, −21.60768222410190261843475534222, −20.67587006040493922605641474074, −19.77677128090977639313844462867, −18.46546320762659150372076571725, −17.99418391723991665621220185275, −16.99512134636278760605270022287, −15.032804679017415641490635580612, −14.43933930543328341986227196769, −13.428125917604598271459515090559, −12.343837272393772704007038519, −11.11226005155192530593249123044, −10.49261358120946042797493986148, −9.45392425445731913887661715575, −8.118848955353684712980202392938, −6.74390748651415169514982435957, −5.18287192514416886111261374755, −4.26709272812152235611576786738, −2.56479556259570197554735125389, −1.8590713408037173097874987460, 0.27510118759088980803606038573, 2.19524729623075723485890802061, 4.23934766360059169765109386065, 5.21404716425151854092859052994, 5.98293518243849897778803468940, 7.606551043237675621375972166390, 8.412721095098794684788563824459, 9.34843676638942628556011823742, 10.72819985586234724257305512183, 12.31189549228837569975293130696, 13.203239011724456720778931032090, 14.10567364028008162111430003353, 15.2039232829073711155115005043, 16.05720352617293657459092657093, 17.399564042972525288462287758969, 17.57531623616164570963174137419, 18.918495988513309252635127090610, 20.34915206568823467311761555629, 21.46934811739634793458923324777, 21.97716188992018722298893389789, 23.56280017925797107305218680695, 24.15686125034108118914299517600, 24.86301393560238898447825922843, 25.79204176820459255651434495329, 26.9086584486080405207779947480

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