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Properties

Label 1-2001-2001.338-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.524 - 0.851i$
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  + (−0.953 − 0.301i)2-s + (0.818 + 0.574i)4-s + (0.917 − 0.396i)5-s + (0.101 − 0.994i)7-s + (−0.607 − 0.794i)8-s + (−0.994 + 0.101i)10-s + (0.359 − 0.933i)11-s + (0.714 − 0.699i)13-s + (−0.396 + 0.917i)14-s + (0.339 + 0.940i)16-s + (0.281 − 0.959i)17-s + (−0.574 + 0.818i)19-s + (0.979 + 0.202i)20-s + (−0.623 + 0.781i)22-s + (0.685 − 0.728i)25-s + (−0.891 + 0.452i)26-s + ⋯
L(s)  = 1  + (−0.953 − 0.301i)2-s + (0.818 + 0.574i)4-s + (0.917 − 0.396i)5-s + (0.101 − 0.994i)7-s + (−0.607 − 0.794i)8-s + (−0.994 + 0.101i)10-s + (0.359 − 0.933i)11-s + (0.714 − 0.699i)13-s + (−0.396 + 0.917i)14-s + (0.339 + 0.940i)16-s + (0.281 − 0.959i)17-s + (−0.574 + 0.818i)19-s + (0.979 + 0.202i)20-s + (−0.623 + 0.781i)22-s + (0.685 − 0.728i)25-s + (−0.891 + 0.452i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.524 - 0.851i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (338, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.524 - 0.851i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6389694687 - 1.144590042i\)
\(L(\frac12)\) \(\approx\) \(0.6389694687 - 1.144590042i\)
\(L(1)\) \(\approx\) \(0.7998029880 - 0.4213325676i\)
\(L(1)\) \(\approx\) \(0.7998029880 - 0.4213325676i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.953 - 0.301i)T \)
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

−20.05529247171904188616798999358, −19.18593833842040345717657712141, −18.66620833717182128601692528182, −17.970861704635865445357052244234, −17.40166899582278348198243880095, −16.80936564665709607594101959702, −15.81653221823448634303285629723, −15.114567350520652890929139789569, −14.62909438941984372064922343591, −13.7859329011128210561578967995, −12.697319348194673226404789982501, −12.00662847798605924875520423052, −11.01518100403486214241280350415, −10.51717266908912131002908343748, −9.53434804995510420444001785956, −9.05413671798281279710031736445, −8.46263276209904145552165764255, −7.27568801746207454696846709377, −6.64852228909970604677502981205, −5.92672325852352126619237586694, −5.30505510549462804191341886976, −4.03065418029524427729013601753, −2.66778074267351181379090229372, −2.02656317404229720458483848461, −1.36980225098711383808022214598, 0.661342016262081243510632463771, 1.26843747228095507512737187777, 2.28753622777795023700007617807, 3.33462741848212224147229683878, 4.04201638076621094781006216966, 5.42408897912906318006995435735, 6.108420377922155156255242922415, 6.95918190744967288364413364729, 7.843856780672038990452152433078, 8.5455659088062821245318711352, 9.30258082217072532446825959681, 9.982818577713243449264813966196, 10.81359596795849164614758547241, 11.13846612168338163470005640343, 12.3806877503520902065883863969, 12.95191302997478221691083907027, 13.87625841868080405819955653555, 14.357354714308872103625041027068, 15.70430207438471231723628857123, 16.359089456723732497568037610451, 16.90857450456735996792278999113, 17.47907182198080422594199866170, 18.28156322402737559706387414604, 18.81092471962835292432383514270, 19.84198651626806395651069197839

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