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Properties

Label 1-667-667.130-r0-0-0
Degree $1$
Conductor $667$
Sign $0.995 - 0.0975i$
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  + (0.998 + 0.0611i)2-s + (−0.396 − 0.917i)3-s + (0.992 + 0.122i)4-s + (0.996 − 0.0815i)5-s + (−0.339 − 0.940i)6-s + (−0.0203 + 0.999i)7-s + (0.983 + 0.182i)8-s + (−0.685 + 0.728i)9-s + (0.999 − 0.0203i)10-s + (−0.925 − 0.377i)11-s + (−0.281 − 0.959i)12-s + (0.452 + 0.891i)13-s + (−0.0815 + 0.996i)14-s + (−0.470 − 0.882i)15-s + (0.970 + 0.242i)16-s + (0.540 + 0.841i)17-s + ⋯
L(s)  = 1  + (0.998 + 0.0611i)2-s + (−0.396 − 0.917i)3-s + (0.992 + 0.122i)4-s + (0.996 − 0.0815i)5-s + (−0.339 − 0.940i)6-s + (−0.0203 + 0.999i)7-s + (0.983 + 0.182i)8-s + (−0.685 + 0.728i)9-s + (0.999 − 0.0203i)10-s + (−0.925 − 0.377i)11-s + (−0.281 − 0.959i)12-s + (0.452 + 0.891i)13-s + (−0.0815 + 0.996i)14-s + (−0.470 − 0.882i)15-s + (0.970 + 0.242i)16-s + (0.540 + 0.841i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(667\)    =    \(23 \cdot 29\)
Sign: $0.995 - 0.0975i$
Analytic conductor: \(3.09753\)
Root analytic conductor: \(3.09753\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{667} (130, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 667,\ (0:\ ),\ 0.995 - 0.0975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.828411055 - 0.1382479282i\)
\(L(\frac12)\) \(\approx\) \(2.828411055 - 0.1382479282i\)
\(L(1)\) \(\approx\) \(1.983477854 - 0.1594410673i\)
\(L(1)\) \(\approx\) \(1.983477854 - 0.1594410673i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.998 + 0.0611i)T \)
3 \( 1 + (-0.396 - 0.917i)T \)
5 \( 1 + (0.996 - 0.0815i)T \)
7 \( 1 + (-0.0203 + 0.999i)T \)
11 \( 1 + (-0.925 - 0.377i)T \)
13 \( 1 + (0.452 + 0.891i)T \)
17 \( 1 + (0.540 + 0.841i)T \)
19 \( 1 + (0.122 - 0.992i)T \)
31 \( 1 + (0.806 + 0.591i)T \)
37 \( 1 + (-0.728 - 0.685i)T \)
41 \( 1 + (-0.989 - 0.142i)T \)
43 \( 1 + (0.806 - 0.591i)T \)
47 \( 1 + (0.781 + 0.623i)T \)
53 \( 1 + (-0.794 + 0.607i)T \)
59 \( 1 + (0.415 - 0.909i)T \)
61 \( 1 + (-0.965 + 0.262i)T \)
67 \( 1 + (-0.377 - 0.925i)T \)
71 \( 1 + (0.818 - 0.574i)T \)
73 \( 1 + (-0.699 - 0.714i)T \)
79 \( 1 + (0.242 + 0.970i)T \)
83 \( 1 + (-0.557 - 0.830i)T \)
89 \( 1 + (-0.470 + 0.882i)T \)
97 \( 1 + (-0.953 + 0.301i)T \)
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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

−22.86790749995254615692146705437, −22.092772015989064273302595382424, −20.92672052616228382287223069018, −20.810722072686927758356626008954, −20.160291842457584408847679113766, −18.63799325407267448430417740213, −17.601085802106408538078581891054, −16.85702580788389997690656842628, −16.10898654701949575563096047176, −15.31987870801191225206349720824, −14.39660014438354249383721940421, −13.70233113523908713425470578561, −12.95776226609981755392947724882, −11.936886643037936724804011950480, −10.85877089359364512985050537970, −10.21115584398241413371028587294, −9.83840576441968865680343869095, −8.148801193733355275933681759020, −7.0672414607054000993133783882, −5.998519940831138687497960101454, −5.3538425244615397838359096317, −4.57772072023767624431535167085, −3.479432593678071130600993085524, −2.6951692357967449633232866031, −1.17748293982125783503171992567, 1.48193272698002925023425549236, 2.278808841348140741851637642576, 3.08764489336809275353939580775, 4.80316680182489690232945763583, 5.569602418526530561819937623924, 6.14512554828767103241392945463, 6.92612796038502536790202680179, 8.106590067113764546984898001403, 9.02502128391404942313074351228, 10.48628985470971893402050334271, 11.18653919481745225442686962607, 12.25466528088055645643260907765, 12.7037724984588689303610612685, 13.70171863124143901588096710471, 14.018456952399620481393976185673, 15.26594008655905191390740543172, 16.10220467065247143200052417311, 17.01812686232417233339189217959, 17.81402375744391050135171339668, 18.76600344899347070137608968901, 19.33636188769017119471085010378, 20.66250380247901234547780207315, 21.42151966818854190645002930484, 21.88662595621636938783117147595, 22.76901201904817710621297319490

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