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Properties

Label 1-2001-2001.470-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.888 + 0.457i$
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  + (0.996 − 0.0815i)2-s + (0.986 − 0.162i)4-s + (−0.591 + 0.806i)5-s + (0.523 + 0.852i)7-s + (0.970 − 0.242i)8-s + (−0.523 + 0.852i)10-s + (0.862 + 0.505i)11-s + (0.101 − 0.994i)13-s + (0.591 + 0.806i)14-s + (0.947 − 0.320i)16-s + (0.959 − 0.281i)17-s + (0.986 − 0.162i)19-s + (−0.452 + 0.891i)20-s + (0.900 + 0.433i)22-s + (−0.301 − 0.953i)25-s + (0.0203 − 0.999i)26-s + ⋯
L(s)  = 1  + (0.996 − 0.0815i)2-s + (0.986 − 0.162i)4-s + (−0.591 + 0.806i)5-s + (0.523 + 0.852i)7-s + (0.970 − 0.242i)8-s + (−0.523 + 0.852i)10-s + (0.862 + 0.505i)11-s + (0.101 − 0.994i)13-s + (0.591 + 0.806i)14-s + (0.947 − 0.320i)16-s + (0.959 − 0.281i)17-s + (0.986 − 0.162i)19-s + (−0.452 + 0.891i)20-s + (0.900 + 0.433i)22-s + (−0.301 − 0.953i)25-s + (0.0203 − 0.999i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.888 + 0.457i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (470, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.888 + 0.457i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.583198810 + 0.8686541931i\)
\(L(\frac12)\) \(\approx\) \(3.583198810 + 0.8686541931i\)
\(L(1)\) \(\approx\) \(2.140885649 + 0.2773610983i\)
\(L(1)\) \(\approx\) \(2.140885649 + 0.2773610983i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.996 - 0.0815i)T \)
5 \( 1 + (-0.591 + 0.806i)T \)
7 \( 1 + (0.523 + 0.852i)T \)
11 \( 1 + (0.862 + 0.505i)T \)
13 \( 1 + (0.101 - 0.994i)T \)
17 \( 1 + (0.959 - 0.281i)T \)
19 \( 1 + (0.986 - 0.162i)T \)
31 \( 1 + (0.979 + 0.202i)T \)
37 \( 1 + (-0.999 - 0.0407i)T \)
41 \( 1 + (-0.654 - 0.755i)T \)
43 \( 1 + (-0.979 + 0.202i)T \)
47 \( 1 + (0.623 - 0.781i)T \)
53 \( 1 + (0.339 - 0.940i)T \)
59 \( 1 + (-0.841 + 0.540i)T \)
61 \( 1 + (-0.768 + 0.639i)T \)
67 \( 1 + (0.862 - 0.505i)T \)
71 \( 1 + (-0.685 - 0.728i)T \)
73 \( 1 + (-0.488 + 0.872i)T \)
79 \( 1 + (0.947 + 0.320i)T \)
83 \( 1 + (0.262 - 0.965i)T \)
89 \( 1 + (-0.794 + 0.607i)T \)
97 \( 1 + (0.917 + 0.396i)T \)
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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

−20.09043613399931057311287116934, −19.41665704310733469867125958374, −18.69539190721289941485171843031, −17.21717291578217829616992204965, −16.86399679927479389146596381880, −16.267679700047980558178423695832, −15.510624039821085717651726920531, −14.58818764679231258269980382371, −13.922949566502854808422469352425, −13.54619299110183311241985567384, −12.41836076539903428134911082817, −11.81169165303889775994234595034, −11.41437683168909441411742429928, −10.44193732498273340459024152232, −9.45820226111387407125809536688, −8.413251597408432243575584846813, −7.751622648502016320995464530146, −7.00019932393512140615133458215, −6.15000037223695673301427424497, −5.175822242574639438297522375851, −4.500171265732034985690166977562, −3.81991535667254523397861170393, −3.19301650687019995500214350544, −1.59312647518952597497067477767, −1.13943809345861870133404250094, 1.179419832692859583017956230107, 2.23150978262720318495417645774, 3.139713885589820852217068014, 3.616154840195622160858021670554, 4.7664702207219172107371847616, 5.406051487987934532487440493699, 6.246274463740420433283314564772, 7.1113688217806688087648276578, 7.72205874412639412057350833348, 8.60377213471815074416332370597, 9.87071260319752229264489033723, 10.48833942909498944430093275545, 11.4537788579998109171276053577, 12.00670551422237444352154709148, 12.32319087568120774089807287224, 13.59230833192647109844291272255, 14.205611560061080683998234424795, 14.9509626368359774112097697304, 15.341003914155456142407464794399, 16.002926840531025884060218702177, 17.03256551615697574457908067428, 17.93734150248668847233304482559, 18.63651098935678995305056068891, 19.42548114517597562888524746577, 20.098899394389639439691864801003

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