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Properties

Label 1-2001-2001.164-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.844 + 0.535i$
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  + (−0.162 + 0.986i)2-s + (−0.947 − 0.320i)4-s + (−0.301 + 0.953i)5-s + (−0.452 − 0.891i)7-s + (0.470 − 0.882i)8-s + (−0.891 − 0.452i)10-s + (0.872 + 0.488i)11-s + (0.979 − 0.202i)13-s + (0.953 − 0.301i)14-s + (0.794 + 0.607i)16-s + (−0.540 + 0.841i)17-s + (0.320 − 0.947i)19-s + (0.591 − 0.806i)20-s + (−0.623 + 0.781i)22-s + (−0.818 − 0.574i)25-s + (0.0407 + 0.999i)26-s + ⋯
L(s)  = 1  + (−0.162 + 0.986i)2-s + (−0.947 − 0.320i)4-s + (−0.301 + 0.953i)5-s + (−0.452 − 0.891i)7-s + (0.470 − 0.882i)8-s + (−0.891 − 0.452i)10-s + (0.872 + 0.488i)11-s + (0.979 − 0.202i)13-s + (0.953 − 0.301i)14-s + (0.794 + 0.607i)16-s + (−0.540 + 0.841i)17-s + (0.320 − 0.947i)19-s + (0.591 − 0.806i)20-s + (−0.623 + 0.781i)22-s + (−0.818 − 0.574i)25-s + (0.0407 + 0.999i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.844 + 0.535i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.844 + 0.535i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.076075093 + 0.3122946473i\)
\(L(\frac12)\) \(\approx\) \(1.076075093 + 0.3122946473i\)
\(L(1)\) \(\approx\) \(0.7881301513 + 0.3590906406i\)
\(L(1)\) \(\approx\) \(0.7881301513 + 0.3590906406i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.162 + 0.986i)T \)
5 \( 1 + (-0.301 + 0.953i)T \)
7 \( 1 + (-0.452 - 0.891i)T \)
11 \( 1 + (0.872 + 0.488i)T \)
13 \( 1 + (0.979 - 0.202i)T \)
17 \( 1 + (-0.540 + 0.841i)T \)
19 \( 1 + (0.320 - 0.947i)T \)
31 \( 1 + (-0.396 - 0.917i)T \)
37 \( 1 + (0.0815 + 0.996i)T \)
41 \( 1 + (-0.989 + 0.142i)T \)
43 \( 1 + (0.396 - 0.917i)T \)
47 \( 1 + (0.974 + 0.222i)T \)
53 \( 1 + (0.768 - 0.639i)T \)
59 \( 1 + (-0.415 - 0.909i)T \)
61 \( 1 + (0.983 - 0.182i)T \)
67 \( 1 + (-0.488 - 0.872i)T \)
71 \( 1 + (-0.0611 - 0.998i)T \)
73 \( 1 + (-0.852 - 0.523i)T \)
79 \( 1 + (-0.607 - 0.794i)T \)
83 \( 1 + (0.862 - 0.505i)T \)
89 \( 1 + (-0.965 + 0.262i)T \)
97 \( 1 + (0.728 + 0.685i)T \)
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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

−19.91622582562303732255779389952, −19.25502811247127259126692912892, −18.556387863172838182497497333446, −17.97101786617074874211216041067, −16.98765269559532578408261195164, −16.259461280827839974348237805308, −15.791013847849893755639491212410, −14.56194958636669239018936592902, −13.78834246644682770154880667999, −13.09331306126453100362177298058, −12.35060625025558137119464222130, −11.77825681504237652555477860642, −11.25589305796046810413073274155, −10.18370531715445170236933123491, −9.24094281503381102795531708088, −8.86916910689758029437603124712, −8.33882084485163964972512685035, −7.16243598027444666205187600146, −5.90769898451806079057067672401, −5.350312176850355258941211571210, −4.22943871713945091232613816524, −3.67541218652402847345149361878, −2.73822278958846350570936646928, −1.647992406074373449098540771061, −0.90577039375377421838074168316, 0.54848001889533784268632892649, 1.77780994897861326338131953890, 3.30633532316850017055837380686, 3.89801167281239163630180313313, 4.605438949435645738952409297890, 5.92167658743408612206672116336, 6.55254384220885084802979203519, 7.04297789470958905661741024926, 7.78749289257498492762478039662, 8.69242597290903668616470334266, 9.4954966134591760726359664212, 10.32194539591665726628265095399, 10.89539557374517132066828604335, 11.83327333392427100736393117834, 13.053520011964836734214291942640, 13.5524299052135517178497078890, 14.247848098133452899663124697, 15.126315316648968609893208954660, 15.47018757843793559718263823335, 16.37150834695194170043589435503, 17.17814291111472121706340974935, 17.64245444954540615565829591145, 18.486639611135989328376395320149, 19.16072703614479263677514844681, 19.82825134918137886921214948222

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