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Properties

Label 1-177-177.104-r1-0-0
Degree $1$
Conductor $177$
Sign $-0.321 - 0.947i$
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  + (−0.468 + 0.883i)2-s + (−0.561 − 0.827i)4-s + (−0.976 + 0.214i)5-s + (0.796 − 0.605i)7-s + (0.994 − 0.108i)8-s + (0.267 − 0.963i)10-s + (0.370 + 0.928i)11-s + (0.0541 + 0.998i)13-s + (0.161 + 0.986i)14-s + (−0.370 + 0.928i)16-s + (−0.796 − 0.605i)17-s + (−0.947 + 0.319i)19-s + (0.725 + 0.687i)20-s + (−0.994 − 0.108i)22-s + (0.856 − 0.515i)23-s + ⋯
L(s)  = 1  + (−0.468 + 0.883i)2-s + (−0.561 − 0.827i)4-s + (−0.976 + 0.214i)5-s + (0.796 − 0.605i)7-s + (0.994 − 0.108i)8-s + (0.267 − 0.963i)10-s + (0.370 + 0.928i)11-s + (0.0541 + 0.998i)13-s + (0.161 + 0.986i)14-s + (−0.370 + 0.928i)16-s + (−0.796 − 0.605i)17-s + (−0.947 + 0.319i)19-s + (0.725 + 0.687i)20-s + (−0.994 − 0.108i)22-s + (0.856 − 0.515i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.321 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.321 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.321 - 0.947i$
Analytic conductor: \(19.0212\)
Root analytic conductor: \(19.0212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (1:\ ),\ -0.321 - 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07409222111 - 0.1033623838i\)
\(L(\frac12)\) \(\approx\) \(0.07409222111 - 0.1033623838i\)
\(L(1)\) \(\approx\) \(0.5665187351 + 0.2045596036i\)
\(L(1)\) \(\approx\) \(0.5665187351 + 0.2045596036i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 \)
good2 \( 1 + (-0.468 + 0.883i)T \)
5 \( 1 + (-0.976 + 0.214i)T \)
7 \( 1 + (0.796 - 0.605i)T \)
11 \( 1 + (0.370 + 0.928i)T \)
13 \( 1 + (0.0541 + 0.998i)T \)
17 \( 1 + (-0.796 - 0.605i)T \)
19 \( 1 + (-0.947 + 0.319i)T \)
23 \( 1 + (0.856 - 0.515i)T \)
29 \( 1 + (-0.468 - 0.883i)T \)
31 \( 1 + (-0.947 - 0.319i)T \)
37 \( 1 + (-0.994 - 0.108i)T \)
41 \( 1 + (0.856 + 0.515i)T \)
43 \( 1 + (-0.370 + 0.928i)T \)
47 \( 1 + (-0.976 - 0.214i)T \)
53 \( 1 + (-0.267 - 0.963i)T \)
61 \( 1 + (0.468 - 0.883i)T \)
67 \( 1 + (-0.994 + 0.108i)T \)
71 \( 1 + (-0.976 - 0.214i)T \)
73 \( 1 + (-0.161 - 0.986i)T \)
79 \( 1 + (-0.725 - 0.687i)T \)
83 \( 1 + (-0.647 + 0.762i)T \)
89 \( 1 + (-0.468 - 0.883i)T \)
97 \( 1 + (-0.161 + 0.986i)T \)
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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

−27.529510018889252996538923222822, −26.968731088669988738514309034285, −25.71781850289329958549445715555, −24.53756425637454701462605439358, −23.58863408852925127733173956682, −22.3641441519104194080662438800, −21.55492172457995098645677451022, −20.56178824446956324530764356813, −19.61363304933610416664102540912, −18.92657534009093716806788545863, −17.838518879587429058313046641355, −16.92992726150291642664219360996, −15.68149553733381773371957136173, −14.66627679849906041788969248838, −13.15430658484602705249349472134, −12.31983532165239217915440372368, −11.1931245501013593806389007282, −10.77191793478254521768844938460, −8.824842558314339919899982221022, −8.53896228051336424424494858287, −7.29194550558583475617741780587, −5.3558286602281036792048081166, −4.09991440499318169709788904523, −2.995044600614856130899758102915, −1.42356608363889455604244828454, 0.05664158325160008183209997064, 1.76200818408201189283178915699, 4.15196413802513982170003553270, 4.74156381096411878296073288241, 6.622836905583063051152241319007, 7.2973532241954621036129148789, 8.31507594210707851490648739067, 9.37936895928922992121781996899, 10.744450186155680585423434632050, 11.57884635292625790731513293832, 13.12411228104957707522691145506, 14.51208564244864381646584712643, 14.91297872200922514141200339969, 16.13816636603987595905326306387, 17.00828528399227701305456676339, 17.95843216026628032077080046057, 18.98639344333662441820927401074, 19.83655949680031919662342457506, 20.890839721035955837474684371680, 22.54681265779476047802779495716, 23.21689665583670174843451178935, 24.04617994500715317263046444699, 24.85526551467306458510785402781, 26.13645590457807581248372457492, 26.78337799637961812234188698585

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