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Properties

Label 1-2001-2001.341-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.467 - 0.884i$
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  + (0.262 − 0.965i)2-s + (−0.862 − 0.505i)4-s + (−0.768 − 0.639i)5-s + (0.818 + 0.574i)7-s + (−0.714 + 0.699i)8-s + (−0.818 + 0.574i)10-s + (0.591 − 0.806i)11-s + (−0.0611 + 0.998i)13-s + (0.768 − 0.639i)14-s + (0.488 + 0.872i)16-s + (0.142 − 0.989i)17-s + (−0.862 − 0.505i)19-s + (0.339 + 0.940i)20-s + (−0.623 − 0.781i)22-s + (0.182 + 0.983i)25-s + (0.947 + 0.320i)26-s + ⋯
L(s)  = 1  + (0.262 − 0.965i)2-s + (−0.862 − 0.505i)4-s + (−0.768 − 0.639i)5-s + (0.818 + 0.574i)7-s + (−0.714 + 0.699i)8-s + (−0.818 + 0.574i)10-s + (0.591 − 0.806i)11-s + (−0.0611 + 0.998i)13-s + (0.768 − 0.639i)14-s + (0.488 + 0.872i)16-s + (0.142 − 0.989i)17-s + (−0.862 − 0.505i)19-s + (0.339 + 0.940i)20-s + (−0.623 − 0.781i)22-s + (0.182 + 0.983i)25-s + (0.947 + 0.320i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.467 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.467 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.467 - 0.884i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.467 - 0.884i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7798270661 - 1.294460066i\)
\(L(\frac12)\) \(\approx\) \(0.7798270661 - 1.294460066i\)
\(L(1)\) \(\approx\) \(0.8763858601 - 0.6314349256i\)
\(L(1)\) \(\approx\) \(0.8763858601 - 0.6314349256i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.262 - 0.965i)T \)
5 \( 1 + (-0.768 - 0.639i)T \)
7 \( 1 + (0.818 + 0.574i)T \)
11 \( 1 + (0.591 - 0.806i)T \)
13 \( 1 + (-0.0611 + 0.998i)T \)
17 \( 1 + (0.142 - 0.989i)T \)
19 \( 1 + (-0.862 - 0.505i)T \)
31 \( 1 + (0.992 + 0.122i)T \)
37 \( 1 + (0.794 + 0.607i)T \)
41 \( 1 + (0.415 + 0.909i)T \)
43 \( 1 + (-0.992 + 0.122i)T \)
47 \( 1 + (-0.222 - 0.974i)T \)
53 \( 1 + (0.742 - 0.670i)T \)
59 \( 1 + (0.959 + 0.281i)T \)
61 \( 1 + (0.101 + 0.994i)T \)
67 \( 1 + (0.591 + 0.806i)T \)
71 \( 1 + (-0.882 - 0.470i)T \)
73 \( 1 + (0.301 - 0.953i)T \)
79 \( 1 + (0.488 - 0.872i)T \)
83 \( 1 + (-0.452 - 0.891i)T \)
89 \( 1 + (0.523 - 0.852i)T \)
97 \( 1 + (0.970 + 0.242i)T \)
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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

−20.13460685887628911190242055296, −19.40068283858139235568790004915, −18.58833183474756844983160114323, −17.726167465733011873075360167775, −17.32169771141175766470741478303, −16.601296119339473032388609257228, −15.54524673541776234913109888301, −15.04111856256926421527374063466, −14.57141490186873377295648851837, −13.90671992766165439858069407342, −12.78798203316559894384946920396, −12.32311040607976844988375993352, −11.33088352806576169156457252498, −10.48874849371474430246160789812, −9.8155949235836103042880312726, −8.50049783096506238634339556376, −8.05582842415036865170807150525, −7.3961932611536494171904378449, −6.662522464206211170029484446415, −5.87631379114182858589898437634, −4.80891611119606439916423652190, −4.09677790847183452620705777306, −3.58805712654987518547631326517, −2.30426380762749743702995721171, −0.89674336711172021664135903980, 0.65540516073392324182158173659, 1.5485035042241323331243953043, 2.51474013189246453424630966581, 3.447661931031020459664477986434, 4.469166262708274745008670178397, 4.74711371800899373698433335418, 5.76122130947268711925394648761, 6.765662067432546027018886938814, 8.01465688499901902994846265104, 8.709706738375770681746016242567, 9.0929904940839451666944696296, 10.10768355891336921119559656305, 11.24700394165208232550031472578, 11.701733808886836311222648090464, 11.90663454196434199845232879668, 13.0864525275002179191299034379, 13.651522722401222064369500024914, 14.60723795906376066358123650944, 15.038900827244828654808850224851, 16.14596668542174307798643427432, 16.79917686349803590851781550731, 17.70436786680296303807125690681, 18.53388871362080513768950772502, 19.1355712988088634896897329300, 19.69130073301489123351645270113

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