Properties

Degree 1
Conductor 167
Sign $0.154 + 0.987i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.752 − 0.658i)2-s + (−0.999 + 0.0378i)3-s + (0.132 + 0.991i)4-s + (−0.843 − 0.537i)5-s + (0.776 + 0.629i)6-s + (−0.521 − 0.853i)7-s + (0.553 − 0.832i)8-s + (0.997 − 0.0756i)9-s + (0.280 + 0.959i)10-s + (−0.982 − 0.188i)11-s + (−0.169 − 0.985i)12-s + (0.351 − 0.936i)13-s + (−0.169 + 0.985i)14-s + (0.862 + 0.505i)15-s + (−0.965 + 0.261i)16-s + (−0.243 + 0.969i)17-s + ⋯
L(s,χ)  = 1  + (−0.752 − 0.658i)2-s + (−0.999 + 0.0378i)3-s + (0.132 + 0.991i)4-s + (−0.843 − 0.537i)5-s + (0.776 + 0.629i)6-s + (−0.521 − 0.853i)7-s + (0.553 − 0.832i)8-s + (0.997 − 0.0756i)9-s + (0.280 + 0.959i)10-s + (−0.982 − 0.188i)11-s + (−0.169 − 0.985i)12-s + (0.351 − 0.936i)13-s + (−0.169 + 0.985i)14-s + (0.862 + 0.505i)15-s + (−0.965 + 0.261i)16-s + (−0.243 + 0.969i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.154 + 0.987i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.154 + 0.987i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(167\)
\( \varepsilon \)  =  $0.154 + 0.987i$
motivic weight  =  \(0\)
character  :  $\chi_{167} (38, \cdot )$
Sato-Tate  :  $\mu(83)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 167,\ (0:\ ),\ 0.154 + 0.987i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.07628786047 + 0.06527251293i$
$L(\frac12,\chi)$  $\approx$  $0.07628786047 + 0.06527251293i$
$L(\chi,1)$  $\approx$  0.3320538944 - 0.1126180162i
$L(1,\chi)$  $\approx$  0.3320538944 - 0.1126180162i

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

−27.438923287734089528654739210874, −26.46280366820777057445367666977, −25.738306208546503480257550043948, −24.293862799367968590548781644554, −23.729576167702152663325100858354, −22.76536951805558083969672896690, −21.97365829241279040126591883330, −20.40018320063559193899070772900, −18.95431229219164870961793441092, −18.5899626862177623261255265691, −17.71834952887481316063807805539, −16.24724732073199727843455078254, −15.884844749385432569896697125062, −15.0017231973701844315573025148, −13.41358492291035986595797569211, −11.91025172013931005630782725011, −11.19957000129877818559390196692, −10.10385701048597986929403682123, −8.959159953370763682410424787375, −7.55415935675886706231733138406, −6.7369260891875085282413257501, −5.70595247980667730856169581144, −4.490461028362494844509408299502, −2.37974867063825295966260682030, −0.132336262702351480365833209540, 1.2468293440255107264389360112, 3.42233661860737690718397323753, 4.39295358919666902877599337659, 6.00952338598893706987953923679, 7.52762246404803229136939865205, 8.195383466573524805306608106819, 9.924092740218492247310601012437, 10.59024911343748290864740941015, 11.53113051087856665563451915010, 12.68133291569369378154179326595, 13.16053459729131209931831717301, 15.53036234019175915972840378889, 16.270393171969972693397593570572, 17.021463626608773221259955400951, 18.05163313090049781422165309269, 19.0116730829981754294231706671, 20.04594371634835682930769763348, 20.816024219604493348344768977617, 21.99087472652711889972447235726, 23.091017702634523068098656759703, 23.68290623741373159502955210567, 25.03125327953025297798748380252, 26.44972828979429050145212884247, 26.977867145452995814951318760420, 28.13809406896573104349250381422

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