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Properties

Label 1-667-667.329-r0-0-0
Degree $1$
Conductor $667$
Sign $0.969 - 0.244i$
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  + (−0.953 − 0.301i)2-s + (0.891 − 0.452i)3-s + (0.818 + 0.574i)4-s + (0.917 − 0.396i)5-s + (−0.986 + 0.162i)6-s + (−0.101 + 0.994i)7-s + (−0.607 − 0.794i)8-s + (0.591 − 0.806i)9-s + (−0.994 + 0.101i)10-s + (−0.359 + 0.933i)11-s + (0.989 + 0.142i)12-s + (0.714 − 0.699i)13-s + (0.396 − 0.917i)14-s + (0.639 − 0.768i)15-s + (0.339 + 0.940i)16-s + (−0.281 + 0.959i)17-s + ⋯
L(s)  = 1  + (−0.953 − 0.301i)2-s + (0.891 − 0.452i)3-s + (0.818 + 0.574i)4-s + (0.917 − 0.396i)5-s + (−0.986 + 0.162i)6-s + (−0.101 + 0.994i)7-s + (−0.607 − 0.794i)8-s + (0.591 − 0.806i)9-s + (−0.994 + 0.101i)10-s + (−0.359 + 0.933i)11-s + (0.989 + 0.142i)12-s + (0.714 − 0.699i)13-s + (0.396 − 0.917i)14-s + (0.639 − 0.768i)15-s + (0.339 + 0.940i)16-s + (−0.281 + 0.959i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(667\)    =    \(23 \cdot 29\)
Sign: $0.969 - 0.244i$
Analytic conductor: \(3.09753\)
Root analytic conductor: \(3.09753\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{667} (329, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 667,\ (0:\ ),\ 0.969 - 0.244i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.519603030 - 0.1888499424i\)
\(L(\frac12)\) \(\approx\) \(1.519603030 - 0.1888499424i\)
\(L(1)\) \(\approx\) \(1.120265676 - 0.1697608359i\)
\(L(1)\) \(\approx\) \(1.120265676 - 0.1697608359i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.953 - 0.301i)T \)
3 \( 1 + (0.891 - 0.452i)T \)
5 \( 1 + (0.917 - 0.396i)T \)
7 \( 1 + (-0.101 + 0.994i)T \)
11 \( 1 + (-0.359 + 0.933i)T \)
13 \( 1 + (0.714 - 0.699i)T \)
17 \( 1 + (-0.281 + 0.959i)T \)
19 \( 1 + (-0.574 + 0.818i)T \)
31 \( 1 + (0.999 + 0.0203i)T \)
37 \( 1 + (-0.806 - 0.591i)T \)
41 \( 1 + (0.755 + 0.654i)T \)
43 \( 1 + (0.999 - 0.0203i)T \)
47 \( 1 + (0.974 + 0.222i)T \)
53 \( 1 + (0.992 - 0.122i)T \)
59 \( 1 + (0.841 + 0.540i)T \)
61 \( 1 + (0.242 - 0.970i)T \)
67 \( 1 + (-0.933 + 0.359i)T \)
71 \( 1 + (-0.996 - 0.0815i)T \)
73 \( 1 + (-0.670 - 0.742i)T \)
79 \( 1 + (-0.940 - 0.339i)T \)
83 \( 1 + (-0.182 + 0.983i)T \)
89 \( 1 + (0.639 + 0.768i)T \)
97 \( 1 + (0.0407 - 0.999i)T \)
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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

−22.84703679288489709745442847123, −21.6744035090669729247440262587, −20.90905916927928602923817814748, −20.46758707648968250387317134162, −19.299238522116061815955153643853, −18.8989708218821878987624765806, −17.88179659687391562358971124387, −17.06004465613322153022311952407, −16.20821025108848731227831840593, −15.623083662901356001751926193516, −14.45881088478325748887905937881, −13.80035945041555471164468282454, −13.32507158596291005560870877193, −11.409522919484830730143097487392, −10.63505135123120577633092248218, −10.11254302492051991567977561543, −9.07933580582496910453004691654, −8.635653104646257923730679897859, −7.390497133980567957186434723168, −6.76118712502422787178726888201, −5.689268497324422980415551769913, −4.402160201825830193298908321354, −3.07441811392174487136474214577, −2.2971892604805851401128717512, −1.046901725841052653878618986519, 1.29448696761315636028937655335, 2.11596288304646734643420390795, 2.76153466165486639495323303678, 4.06917035139496858957043004844, 5.76168017345440797869450998245, 6.44818666913561284632672015084, 7.66712164912148017329636058723, 8.5109238065441641977199028073, 8.98681704801984101022154459588, 9.944056371315429188173523170, 10.5637575822541908646348489168, 12.12300961319677678498438306873, 12.66948820777331536853316036129, 13.278923509746858522474427955601, 14.63091940509537072509478153859, 15.35199086596257364193489031788, 16.141797471810444722414375110454, 17.52399517398646400615434667903, 17.80306118617522693069156174056, 18.6927204159094229760825509404, 19.36053515335800906874529335855, 20.31354775289999834163309955491, 20.98990779298757900875468549953, 21.39520245368531697951254924186, 22.61162014192759921378956780913

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