Properties

Label 1-2001-2001.776-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.932 + 0.361i$
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  + (0.917 + 0.396i)2-s + (0.685 + 0.728i)4-s + (0.0203 + 0.999i)5-s + (0.377 + 0.925i)7-s + (0.339 + 0.940i)8-s + (−0.377 + 0.925i)10-s + (−0.882 − 0.470i)11-s + (0.488 + 0.872i)13-s + (−0.0203 + 0.999i)14-s + (−0.0611 + 0.998i)16-s + (0.142 + 0.989i)17-s + (0.685 + 0.728i)19-s + (−0.714 + 0.699i)20-s + (−0.623 − 0.781i)22-s + (−0.999 + 0.0407i)25-s + (0.101 + 0.994i)26-s + ⋯
L(s)  = 1  + (0.917 + 0.396i)2-s + (0.685 + 0.728i)4-s + (0.0203 + 0.999i)5-s + (0.377 + 0.925i)7-s + (0.339 + 0.940i)8-s + (−0.377 + 0.925i)10-s + (−0.882 − 0.470i)11-s + (0.488 + 0.872i)13-s + (−0.0203 + 0.999i)14-s + (−0.0611 + 0.998i)16-s + (0.142 + 0.989i)17-s + (0.685 + 0.728i)19-s + (−0.714 + 0.699i)20-s + (−0.623 − 0.781i)22-s + (−0.999 + 0.0407i)25-s + (0.101 + 0.994i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.932 + 0.361i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (776, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.932 + 0.361i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5366005662 + 2.866207383i\)
\(L(\frac12)\) \(\approx\) \(0.5366005662 + 2.866207383i\)
\(L(1)\) \(\approx\) \(1.393580308 + 1.223362787i\)
\(L(1)\) \(\approx\) \(1.393580308 + 1.223362787i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.917 + 0.396i)T \)
5 \( 1 + (0.0203 + 0.999i)T \)
7 \( 1 + (0.377 + 0.925i)T \)
11 \( 1 + (-0.882 - 0.470i)T \)
13 \( 1 + (0.488 + 0.872i)T \)
17 \( 1 + (0.142 + 0.989i)T \)
19 \( 1 + (0.685 + 0.728i)T \)
31 \( 1 + (0.523 - 0.852i)T \)
37 \( 1 + (-0.979 + 0.202i)T \)
41 \( 1 + (0.415 - 0.909i)T \)
43 \( 1 + (-0.523 - 0.852i)T \)
47 \( 1 + (-0.222 - 0.974i)T \)
53 \( 1 + (0.986 - 0.162i)T \)
59 \( 1 + (0.959 - 0.281i)T \)
61 \( 1 + (0.947 + 0.320i)T \)
67 \( 1 + (-0.882 + 0.470i)T \)
71 \( 1 + (0.591 - 0.806i)T \)
73 \( 1 + (-0.557 + 0.830i)T \)
79 \( 1 + (-0.0611 - 0.998i)T \)
83 \( 1 + (0.970 + 0.242i)T \)
89 \( 1 + (0.992 + 0.122i)T \)
97 \( 1 + (-0.452 - 0.891i)T \)
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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

−20.0721359195268580890089005131, −19.21549950325319098106613258042, −17.97568734643267613047403304943, −17.63838440720484960523491446350, −16.3385311152054604494806163141, −16.02409562812266536512981126014, −15.2461400779879489392572307844, −14.30856922330504966503143437493, −13.43333700066609206774111484753, −13.25004617222870947907624388232, −12.35383194636988710186126299793, −11.614294163468662013487700142312, −10.82236258947286860346055804008, −10.116363626537302086964836380182, −9.39988405153032878620370638850, −8.19828096946809394171354999685, −7.51618372809126714747786027374, −6.70516434818247907388700317501, −5.47811222431552048923721446356, −5.02261940070331681189406671606, −4.41267190303156956563644842201, −3.39358942862032993372140461629, −2.581325089562889855390112381311, −1.373623093361221541173074559187, −0.71165723374716230114142233782, 1.82555110368531380393177024609, 2.41684051011074673544386021480, 3.40016425327885982271440537018, 3.98650968428647575568195840028, 5.25468908300598776230145393688, 5.766446065243043991677460214035, 6.47354004427243117807993600935, 7.32942885240360573602569490600, 8.15046228803751081990114501498, 8.75045079746788632327876656572, 10.11243951963611246112585831616, 10.7895435204320524634613510488, 11.67089310424287408198781034854, 12.012555923545646596915730157760, 13.13787856844658492498217052862, 13.76130406034860034104665781271, 14.4706851573940845826210889142, 15.09319572189111919104700734273, 15.69954185516147147590335527264, 16.369898053067878926572714926015, 17.321781907668098641423009150957, 18.144153059239352114464066371257, 18.798767078909304293725826347585, 19.36992885777541778242416340026, 20.795427073434884632834456248658

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