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Properties

Label 1-2001-2001.1250-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.955 - 0.295i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.955 - 0.295i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1250, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.955 - 0.295i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08694474530 + 0.5755502013i\)
\(L(\frac12)\) \(\approx\) \(-0.08694474530 + 0.5755502013i\)
\(L(1)\) \(\approx\) \(0.6790424692 + 0.4256821991i\)
\(L(1)\) \(\approx\) \(0.6790424692 + 0.4256821991i\)

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

−19.50228686461955223465442880657, −18.960341333194556850369497752285, −18.20009726753749036464807958212, −17.84122015982305704132914561056, −16.53091034467864204649128494440, −15.99279088059073238747616780533, −14.97245346616186019950475654615, −14.09944796703754356983436918025, −13.73338430886812312817810918532, −12.915605409830398973143118211047, −12.13936810818453679532516852302, −11.15246447277241768130584735325, −10.73226163315814272700158293784, −10.168963253455715993147578738605, −9.38448140069557710924754264608, −8.20275360857720138462152978404, −7.68739253841741869612761098691, −6.552242352946030524041906397734, −5.80728221072313151647576154123, −4.79422010646101349002973102083, −3.697625369366960395430900324641, −3.35607088419307958376169766597, −2.53864749402628623963359690586, −1.32564738308858356201103993970, −0.21684866312563824311613779035, 1.16556179930494441529863000571, 2.518250856907524035332624432489, 3.6132483946015691327568124999, 4.43979171800540633250291843001, 5.20936776977419993975473394680, 5.89228524989924985270296232284, 6.59330601850732642817188290739, 7.729497182000851417385873913887, 8.27768309694752340073235938453, 8.968672862347194695278472116537, 9.57408743346669491657955815545, 10.58645807586914047825771993117, 11.827341352657593274176375583396, 12.41969481617261902920504803305, 13.268394033663729200099936833197, 13.43367897046045959411756362528, 14.897529218008383785423687928258, 15.314394141632720786186816188319, 15.92981124120833404751786951683, 16.540280460713072140964407617256, 17.299063279366680957302341809047, 18.071877197864118788804050968037, 18.7895302573141158514908160423, 19.40914912672479757450411498500, 20.48267209757857609505423761741

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