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Properties

Label 1-167-167.22-r0-0-0
Degree $1$
Conductor $167$
Sign $0.154 - 0.987i$
Analytic cond. $0.775544$
Root an. cond. $0.775544$
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  + (−0.752 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(167\)
Sign: $0.154 - 0.987i$
Analytic conductor: \(0.775544\)
Root analytic conductor: \(0.775544\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{167} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 167,\ (0:\ ),\ 0.154 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07628786047 - 0.06527251293i\)
\(L(\frac12)\) \(\approx\) \(0.07628786047 - 0.06527251293i\)
\(L(1)\) \(\approx\) \(0.3320538944 + 0.1126180162i\)
\(L(1)\) \(\approx\) \(0.3320538944 + 0.1126180162i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad167 \( 1 \)
good2 \( 1 + (-0.752 + 0.658i)T \)
3 \( 1 + (-0.999 - 0.0378i)T \)
5 \( 1 + (-0.843 + 0.537i)T \)
7 \( 1 + (-0.521 + 0.853i)T \)
11 \( 1 + (-0.982 + 0.188i)T \)
13 \( 1 + (0.351 + 0.936i)T \)
17 \( 1 + (-0.243 - 0.969i)T \)
19 \( 1 + (0.0567 - 0.998i)T \)
23 \( 1 + (-0.942 - 0.334i)T \)
29 \( 1 + (-0.942 + 0.334i)T \)
31 \( 1 + (0.672 - 0.739i)T \)
37 \( 1 + (0.997 - 0.0756i)T \)
41 \( 1 + (-0.0944 - 0.995i)T \)
43 \( 1 + (-0.644 - 0.764i)T \)
47 \( 1 + (0.898 + 0.438i)T \)
53 \( 1 + (-0.584 - 0.811i)T \)
59 \( 1 + (-0.243 + 0.969i)T \)
61 \( 1 + (0.974 + 0.225i)T \)
67 \( 1 + (-0.843 - 0.537i)T \)
71 \( 1 + (-0.914 + 0.404i)T \)
73 \( 1 + (-0.965 + 0.261i)T \)
79 \( 1 + (-0.700 - 0.713i)T \)
83 \( 1 + (-0.752 - 0.658i)T \)
89 \( 1 + (0.862 + 0.505i)T \)
97 \( 1 + (0.672 + 0.739i)T \)
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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

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

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