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Properties

Label 1-2001-2001.95-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.875 + 0.483i$
Analytic cond. $9.29260$
Root an. cond. $9.29260$
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  + (0.925 + 0.377i)2-s + (0.714 + 0.699i)4-s + (−0.862 − 0.505i)5-s + (0.794 + 0.607i)7-s + (0.396 + 0.917i)8-s + (−0.607 − 0.794i)10-s + (−0.162 − 0.986i)11-s + (0.768 + 0.639i)13-s + (0.505 + 0.862i)14-s + (0.0203 + 0.999i)16-s + (0.540 − 0.841i)17-s + (0.699 − 0.714i)19-s + (−0.262 − 0.965i)20-s + (0.222 − 0.974i)22-s + (0.488 + 0.872i)25-s + (0.470 + 0.882i)26-s + ⋯
L(s)  = 1  + (0.925 + 0.377i)2-s + (0.714 + 0.699i)4-s + (−0.862 − 0.505i)5-s + (0.794 + 0.607i)7-s + (0.396 + 0.917i)8-s + (−0.607 − 0.794i)10-s + (−0.162 − 0.986i)11-s + (0.768 + 0.639i)13-s + (0.505 + 0.862i)14-s + (0.0203 + 0.999i)16-s + (0.540 − 0.841i)17-s + (0.699 − 0.714i)19-s + (−0.262 − 0.965i)20-s + (0.222 − 0.974i)22-s + (0.488 + 0.872i)25-s + (0.470 + 0.882i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.875 + 0.483i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.875 + 0.483i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.957445796 + 0.7626504894i\)
\(L(\frac12)\) \(\approx\) \(2.957445796 + 0.7626504894i\)
\(L(1)\) \(\approx\) \(1.829964773 + 0.3794722596i\)
\(L(1)\) \(\approx\) \(1.829964773 + 0.3794722596i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.925 + 0.377i)T \)
5 \( 1 + (-0.862 - 0.505i)T \)
7 \( 1 + (0.794 + 0.607i)T \)
11 \( 1 + (-0.162 - 0.986i)T \)
13 \( 1 + (0.768 + 0.639i)T \)
17 \( 1 + (0.540 - 0.841i)T \)
19 \( 1 + (0.699 - 0.714i)T \)
31 \( 1 + (-0.983 + 0.182i)T \)
37 \( 1 + (-0.830 - 0.557i)T \)
41 \( 1 + (0.989 - 0.142i)T \)
43 \( 1 + (0.983 + 0.182i)T \)
47 \( 1 + (-0.433 - 0.900i)T \)
53 \( 1 + (0.452 + 0.891i)T \)
59 \( 1 + (-0.415 - 0.909i)T \)
61 \( 1 + (0.806 - 0.591i)T \)
67 \( 1 + (-0.986 - 0.162i)T \)
71 \( 1 + (0.742 - 0.670i)T \)
73 \( 1 + (0.320 - 0.947i)T \)
79 \( 1 + (0.999 + 0.0203i)T \)
83 \( 1 + (-0.996 + 0.0815i)T \)
89 \( 1 + (0.0407 + 0.999i)T \)
97 \( 1 + (0.359 + 0.933i)T \)
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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

−19.98981155811499521917341971929, −19.448744090669923958010128935563, −18.48271799292735098308233699160, −17.92193329371984219433974956169, −16.858391492194718782459926244, −15.942568684517400687928455068201, −15.37325062084734243785229831377, −14.54097486133213801565988559146, −14.30467124953098671537992906185, −13.17411955656018896030318688793, −12.52459683903938926749907944409, −11.80072466193688014195008453235, −11.0598578827316922699577439474, −10.502973133356172662638543564377, −9.86698045551885574921505996143, −8.4176159429577518704998197258, −7.57866797517364276930683267905, −7.180642717374011231942919990130, −6.04590422280221225258373713477, −5.28065321409820987154077889345, −4.29199429639066380155075159908, −3.80733793250313106276668787656, −3.00869859806987298749005006897, −1.85646559272209443732549448672, −1.0289227523531898395903101411, 0.96624917756750414819286992786, 2.1497275869618281779444242106, 3.222922414757988498436207973761, 3.83586272039439317855752189977, 4.86140373497232857987696952520, 5.32451631732586432557383889910, 6.165626178036844374659305425833, 7.2824919190008235365161749729, 7.803291476741066501087506464163, 8.69264504633442101262409340859, 9.15664243196338865452144825896, 10.99293340913942512598651894677, 11.203126682672914961731255412584, 11.986893307193769506512290052909, 12.56003441184126634938026517698, 13.61074272196341130290570503199, 14.06678751880959741823099794252, 14.8970912283067510193861062081, 15.75605307079759654901148551142, 16.108654009281171965173217414816, 16.74134627372072275890632957504, 17.8036335364378223026113178547, 18.55784097396461283358486116824, 19.3752185534228866434094016858, 20.234909232491049443493572678454

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