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Properties

Label 1-667-667.148-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: $\chi_{667} (148, \cdot )$
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

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

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