Properties

Label 1-177-177.71-r1-0-0
Degree $1$
Conductor $177$
Sign $0.734 + 0.678i$
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  + (−0.796 + 0.605i)2-s + (0.267 − 0.963i)4-s + (0.725 − 0.687i)5-s + (0.647 + 0.762i)7-s + (0.370 + 0.928i)8-s + (−0.161 + 0.986i)10-s + (0.856 − 0.515i)11-s + (−0.561 + 0.827i)13-s + (−0.976 − 0.214i)14-s + (−0.856 − 0.515i)16-s + (−0.647 + 0.762i)17-s + (0.907 + 0.419i)19-s + (−0.468 − 0.883i)20-s + (−0.370 + 0.928i)22-s + (0.947 − 0.319i)23-s + ⋯
L(s)  = 1  + (−0.796 + 0.605i)2-s + (0.267 − 0.963i)4-s + (0.725 − 0.687i)5-s + (0.647 + 0.762i)7-s + (0.370 + 0.928i)8-s + (−0.161 + 0.986i)10-s + (0.856 − 0.515i)11-s + (−0.561 + 0.827i)13-s + (−0.976 − 0.214i)14-s + (−0.856 − 0.515i)16-s + (−0.647 + 0.762i)17-s + (0.907 + 0.419i)19-s + (−0.468 − 0.883i)20-s + (−0.370 + 0.928i)22-s + (0.947 − 0.319i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.734 + 0.678i$
Analytic conductor: \(19.0212\)
Root analytic conductor: \(19.0212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (1:\ ),\ 0.734 + 0.678i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.488758370 + 0.5820332816i\)
\(L(\frac12)\) \(\approx\) \(1.488758370 + 0.5820332816i\)
\(L(1)\) \(\approx\) \(0.9658534322 + 0.2307647412i\)
\(L(1)\) \(\approx\) \(0.9658534322 + 0.2307647412i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 \)
good2 \( 1 + (-0.796 + 0.605i)T \)
5 \( 1 + (0.725 - 0.687i)T \)
7 \( 1 + (0.647 + 0.762i)T \)
11 \( 1 + (0.856 - 0.515i)T \)
13 \( 1 + (-0.561 + 0.827i)T \)
17 \( 1 + (-0.647 + 0.762i)T \)
19 \( 1 + (0.907 + 0.419i)T \)
23 \( 1 + (0.947 - 0.319i)T \)
29 \( 1 + (-0.796 - 0.605i)T \)
31 \( 1 + (0.907 - 0.419i)T \)
37 \( 1 + (-0.370 + 0.928i)T \)
41 \( 1 + (0.947 + 0.319i)T \)
43 \( 1 + (-0.856 - 0.515i)T \)
47 \( 1 + (0.725 + 0.687i)T \)
53 \( 1 + (0.161 + 0.986i)T \)
61 \( 1 + (0.796 - 0.605i)T \)
67 \( 1 + (-0.370 - 0.928i)T \)
71 \( 1 + (0.725 + 0.687i)T \)
73 \( 1 + (0.976 + 0.214i)T \)
79 \( 1 + (0.468 + 0.883i)T \)
83 \( 1 + (0.994 + 0.108i)T \)
89 \( 1 + (-0.796 - 0.605i)T \)
97 \( 1 + (0.976 - 0.214i)T \)
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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

−26.9578278083512929516396422480, −26.44092710801620115477506202835, −25.19557512311636558259924687630, −24.61198771284632034984522262330, −22.80233732149080521609150771682, −22.17502944273667378813700266596, −21.03735222418812207060509157032, −20.20348129386443721651173041459, −19.37303537021185242392621269873, −17.87476276684850133311410595594, −17.73553213942049193086650427715, −16.676144236122653105016053435937, −15.19232966552416835201567232875, −14.05818307800188676926644579548, −13.05745066167401456270516209792, −11.6819195188761917484058026571, −10.8462254404288546498744683, −9.90459951533065418480894697946, −9.02817525927263822331717188266, −7.45570507389490468557810699915, −6.89706438676564860468386162461, −5.00511454056876448474146633013, −3.468033568853178458937469032562, −2.23170059087582142630563430791, −0.932713878149190438700384370366, 1.14692244114607732945063080367, 2.203330557436917779409846370412, 4.59485314734333905009310209003, 5.67157066850319026277171339119, 6.5987647377907205821637671885, 8.1044967392012393370069054642, 8.99397653084703785207816705240, 9.64985000349424423757514187802, 11.12043715465959730207552692512, 12.10586486691719166861105223134, 13.67841840693772967822405043609, 14.55145924170576234067787993587, 15.56792606998604161001271365227, 16.87276816472612643400258946094, 17.21599353688534085495385004321, 18.421876429331781018177156701928, 19.25963975890353473751037742286, 20.42000209656827485564107293751, 21.39928236102852918611087721709, 22.419294469618176599531981830073, 24.0728855827579695643256863341, 24.51532293276346403344471329727, 25.16185589038553138712842631485, 26.378358013848803514514047291643, 27.1922792462585414490257506830

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