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Properties

Label 1-2001-2001.305-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.245 + 0.969i$
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  + (0.806 + 0.591i)2-s + (0.301 + 0.953i)4-s + (−0.979 − 0.202i)5-s + (0.742 + 0.670i)7-s + (−0.320 + 0.947i)8-s + (−0.670 − 0.742i)10-s + (0.983 − 0.182i)11-s + (0.377 − 0.925i)13-s + (0.202 + 0.979i)14-s + (−0.818 + 0.574i)16-s + (−0.989 + 0.142i)17-s + (0.953 − 0.301i)19-s + (−0.101 − 0.994i)20-s + (0.900 + 0.433i)22-s + (0.917 + 0.396i)25-s + (0.852 − 0.523i)26-s + ⋯
L(s)  = 1  + (0.806 + 0.591i)2-s + (0.301 + 0.953i)4-s + (−0.979 − 0.202i)5-s + (0.742 + 0.670i)7-s + (−0.320 + 0.947i)8-s + (−0.670 − 0.742i)10-s + (0.983 − 0.182i)11-s + (0.377 − 0.925i)13-s + (0.202 + 0.979i)14-s + (−0.818 + 0.574i)16-s + (−0.989 + 0.142i)17-s + (0.953 − 0.301i)19-s + (−0.101 − 0.994i)20-s + (0.900 + 0.433i)22-s + (0.917 + 0.396i)25-s + (0.852 − 0.523i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.245 + 0.969i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.245 + 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.074365482 + 1.614857303i\)
\(L(\frac12)\) \(\approx\) \(2.074365482 + 1.614857303i\)
\(L(1)\) \(\approx\) \(1.502756689 + 0.6930422120i\)
\(L(1)\) \(\approx\) \(1.502756689 + 0.6930422120i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.806 + 0.591i)T \)
5 \( 1 + (-0.979 - 0.202i)T \)
7 \( 1 + (0.742 + 0.670i)T \)
11 \( 1 + (0.983 - 0.182i)T \)
13 \( 1 + (0.377 - 0.925i)T \)
17 \( 1 + (-0.989 + 0.142i)T \)
19 \( 1 + (0.953 - 0.301i)T \)
31 \( 1 + (0.699 - 0.714i)T \)
37 \( 1 + (0.891 + 0.452i)T \)
41 \( 1 + (0.909 + 0.415i)T \)
43 \( 1 + (-0.699 - 0.714i)T \)
47 \( 1 + (0.781 + 0.623i)T \)
53 \( 1 + (0.0611 - 0.998i)T \)
59 \( 1 + (0.959 - 0.281i)T \)
61 \( 1 + (-0.122 - 0.992i)T \)
67 \( 1 + (-0.182 + 0.983i)T \)
71 \( 1 + (-0.999 + 0.0407i)T \)
73 \( 1 + (-0.359 + 0.933i)T \)
79 \( 1 + (0.574 - 0.818i)T \)
83 \( 1 + (0.768 + 0.639i)T \)
89 \( 1 + (-0.940 - 0.339i)T \)
97 \( 1 + (0.999 - 0.0203i)T \)
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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

−19.76805092010826468438177513273, −19.49387791430428773413166754861, −18.45309543952047877000810280264, −17.83835682171726765437370964796, −16.668218240127099136071169985292, −16.05331894613611495147726892275, −15.209806016338847258283151755, −14.51514852813796733744820925249, −13.97822164049149974909267315394, −13.29879333636674980005315351378, −12.118287367585766204326049870685, −11.73896075719408377196035386456, −11.12484392956596487394689224864, −10.489788468227554692289612222786, −9.40935489235975655621351660428, −8.66442318226192785383161816630, −7.49295199330598944399749406928, −6.91922031033320839664313642843, −6.12402902881708760003429187018, −4.88567641397883328723145696854, −4.23553352474450217566909527318, −3.84060454197849840431341239513, −2.785417862808258123869813245462, −1.66123104156223602078886478055, −0.88719480418151956503357228561, 1.00797342042357985466366348544, 2.37388903827885805194791904335, 3.25975734080941663552290456027, 4.10392906714734198947446206426, 4.747313515974254525657681135261, 5.582661143709301409448633061151, 6.38223228910185391690552287611, 7.275469140374375586499644142441, 8.0761192186182221284282833114, 8.540072929302248738687380339702, 9.370605493059632221455243895163, 10.88578531836085400136634446789, 11.56278383234145003835711041635, 11.878637616395959956644524982563, 12.83753891793075193218678335041, 13.481802313197503456691758487720, 14.48772868036287446682224086781, 14.98868879540852233886481432375, 15.67128151881811668291756906406, 16.12900867021135327407642042660, 17.176404321059959650180795462432, 17.715394737980240152302380736588, 18.55222955285683278618276285314, 19.55683934196574870772148953217, 20.31478546677825920498747558789

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