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Properties

Label 1-667-667.439-r0-0-0
Degree $1$
Conductor $667$
Sign $-0.962 + 0.272i$
Analytic cond. $3.09753$
Root an. cond. $3.09753$
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  + (−0.0203 + 0.999i)2-s + (0.377 − 0.925i)3-s + (−0.999 − 0.0407i)4-s + (−0.523 − 0.852i)5-s + (0.917 + 0.396i)6-s + (−0.862 − 0.505i)7-s + (0.0611 − 0.998i)8-s + (−0.714 − 0.699i)9-s + (0.862 − 0.505i)10-s + (−0.794 − 0.607i)11-s + (−0.415 + 0.909i)12-s + (−0.933 − 0.359i)13-s + (0.523 − 0.852i)14-s + (−0.986 + 0.162i)15-s + (0.996 + 0.0815i)16-s + (0.654 + 0.755i)17-s + ⋯
L(s)  = 1  + (−0.0203 + 0.999i)2-s + (0.377 − 0.925i)3-s + (−0.999 − 0.0407i)4-s + (−0.523 − 0.852i)5-s + (0.917 + 0.396i)6-s + (−0.862 − 0.505i)7-s + (0.0611 − 0.998i)8-s + (−0.714 − 0.699i)9-s + (0.862 − 0.505i)10-s + (−0.794 − 0.607i)11-s + (−0.415 + 0.909i)12-s + (−0.933 − 0.359i)13-s + (0.523 − 0.852i)14-s + (−0.986 + 0.162i)15-s + (0.996 + 0.0815i)16-s + (0.654 + 0.755i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(667\)    =    \(23 \cdot 29\)
Sign: $-0.962 + 0.272i$
Analytic conductor: \(3.09753\)
Root analytic conductor: \(3.09753\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{667} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 667,\ (0:\ ),\ -0.962 + 0.272i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02547575832 - 0.1835349137i\)
\(L(\frac12)\) \(\approx\) \(0.02547575832 - 0.1835349137i\)
\(L(1)\) \(\approx\) \(0.6471661923 - 0.09940274854i\)
\(L(1)\) \(\approx\) \(0.6471661923 - 0.09940274854i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.0203 + 0.999i)T \)
3 \( 1 + (0.377 - 0.925i)T \)
5 \( 1 + (-0.523 - 0.852i)T \)
7 \( 1 + (-0.862 - 0.505i)T \)
11 \( 1 + (-0.794 - 0.607i)T \)
13 \( 1 + (-0.933 - 0.359i)T \)
17 \( 1 + (0.654 + 0.755i)T \)
19 \( 1 + (0.999 + 0.0407i)T \)
31 \( 1 + (-0.742 - 0.670i)T \)
37 \( 1 + (0.714 + 0.699i)T \)
41 \( 1 + (-0.841 - 0.540i)T \)
43 \( 1 + (-0.742 + 0.670i)T \)
47 \( 1 + (0.222 - 0.974i)T \)
53 \( 1 + (-0.301 + 0.953i)T \)
59 \( 1 + (-0.142 + 0.989i)T \)
61 \( 1 + (0.818 - 0.574i)T \)
67 \( 1 + (0.794 - 0.607i)T \)
71 \( 1 + (-0.979 + 0.202i)T \)
73 \( 1 + (-0.262 + 0.965i)T \)
79 \( 1 + (-0.996 + 0.0815i)T \)
83 \( 1 + (0.947 + 0.320i)T \)
89 \( 1 + (-0.986 - 0.162i)T \)
97 \( 1 + (-0.101 - 0.994i)T \)
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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

−22.885513116788458451545570413911, −22.12716282157052044544319732218, −21.79119036758824132393614865454, −20.68056602006212218511177477503, −20.021859986691056883748551313190, −19.27598211176291785705944268049, −18.600236101817195323060425595272, −17.779964375660119582658730831272, −16.46136932664303783533981907113, −15.762894994407341782477752203762, −14.78730355600555770513156022550, −14.233309171653964020481978076663, −13.15268445888127372295700456232, −12.11544530469311508302684743029, −11.47546091497311505289313347570, −10.41144602016723141913358494856, −9.82992852814794815421578858278, −9.260117763586366830987918840450, −8.026912923226855420230077411335, −7.14690639280560128103545127518, −5.505293545779123961222525269353, −4.75732215102348383511408988668, −3.57384872661034478856131814503, −2.957568058828247260093226044658, −2.25726607332086445143037036233, 0.09618186443557739244424937854, 1.15484415670798751144885589555, 3.009989266448262820267244838, 3.86223771436092644207873556341, 5.21376684986105509829207562727, 5.91914374293398476257340482305, 7.08499999245200543061652119349, 7.73308710702003457301431523098, 8.32695737936134473521039824786, 9.339259395756895777095646989327, 10.16542417057155755038132241475, 11.78172338111205576915184343832, 12.74086612881668966144991473883, 13.13209613983685027787659546730, 13.93790432464815994465160972742, 14.95699498017117270595904028227, 15.760852362150922089764038717442, 16.7308368842088316488711237599, 17.08017148915432176736044812108, 18.323255453459213956554352165531, 18.94748507866541288830872838686, 19.76346087679114609821486041159, 20.39943047541131063720322180847, 21.724710187259489037558268787055, 22.715431539688493321107263661960

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