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Properties

Label 1-2001-2001.998-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.258 - 0.966i$
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  + (−0.540 − 0.841i)2-s + (−0.415 + 0.909i)4-s + (−0.959 + 0.281i)5-s + (−0.654 − 0.755i)7-s + (0.989 − 0.142i)8-s + (0.755 + 0.654i)10-s + (0.540 − 0.841i)11-s + (0.654 − 0.755i)13-s + (−0.281 + 0.959i)14-s + (−0.654 − 0.755i)16-s + (0.909 − 0.415i)17-s + (0.909 + 0.415i)19-s + (0.142 − 0.989i)20-s − 22-s + (0.841 − 0.540i)25-s + (−0.989 − 0.142i)26-s + ⋯
L(s)  = 1  + (−0.540 − 0.841i)2-s + (−0.415 + 0.909i)4-s + (−0.959 + 0.281i)5-s + (−0.654 − 0.755i)7-s + (0.989 − 0.142i)8-s + (0.755 + 0.654i)10-s + (0.540 − 0.841i)11-s + (0.654 − 0.755i)13-s + (−0.281 + 0.959i)14-s + (−0.654 − 0.755i)16-s + (0.909 − 0.415i)17-s + (0.909 + 0.415i)19-s + (0.142 − 0.989i)20-s − 22-s + (0.841 − 0.540i)25-s + (−0.989 − 0.142i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.258 - 0.966i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (998, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.258 - 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5874571230 - 0.7651272626i\)
\(L(\frac12)\) \(\approx\) \(0.5874571230 - 0.7651272626i\)
\(L(1)\) \(\approx\) \(0.6408295049 - 0.3497745806i\)
\(L(1)\) \(\approx\) \(0.6408295049 - 0.3497745806i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.540 - 0.841i)T \)
5 \( 1 + (-0.959 + 0.281i)T \)
7 \( 1 + (-0.654 - 0.755i)T \)
11 \( 1 + (0.540 - 0.841i)T \)
13 \( 1 + (0.654 - 0.755i)T \)
17 \( 1 + (0.909 - 0.415i)T \)
19 \( 1 + (0.909 + 0.415i)T \)
31 \( 1 + (0.989 - 0.142i)T \)
37 \( 1 + (-0.281 + 0.959i)T \)
41 \( 1 + (-0.281 - 0.959i)T \)
43 \( 1 + (-0.989 - 0.142i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.654 + 0.755i)T \)
59 \( 1 + (0.654 - 0.755i)T \)
61 \( 1 + (0.989 - 0.142i)T \)
67 \( 1 + (-0.841 + 0.540i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (-0.909 - 0.415i)T \)
79 \( 1 + (-0.755 - 0.654i)T \)
83 \( 1 + (0.959 + 0.281i)T \)
89 \( 1 + (0.989 + 0.142i)T \)
97 \( 1 + (0.281 + 0.959i)T \)
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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

−19.7382551596377887281330583709, −19.44414489873775278615092919225, −18.622029090580110240873926765585, −18.08666780564741142057399529523, −17.10169340674175707804359006876, −16.32354941590502748671258016565, −15.979403712256853934225998811865, −15.13243892618404232568614598851, −14.69239986534860987241076300458, −13.6768465059440253356352409943, −12.830078464548967521247760297154, −11.90807887684974750652863991746, −11.45928438432201255426267990280, −10.17304887080294486174486530647, −9.59718651269779752045489165493, −8.76555770309242229574410312982, −8.28952514737803535343029384744, −7.23424481151028586874372931916, −6.767120038782547875153278782530, −5.82837804871033432370690191400, −5.0031361528463134151146712149, −4.13345691327510954317987922768, −3.27991063359869965320909293033, −1.87303453209081306378938235749, −0.87632092098673468307087101902, 0.636524781180682298464293726305, 1.19372175099730137145917601466, 2.86615710895773537229957075866, 3.4300175368330962195786849624, 3.8423206150354688760271495403, 4.98658895418549271687241857597, 6.217125298538465991843481674546, 7.1482125085884322954890690807, 7.8636459517558625707037176483, 8.4458939807130354664475911104, 9.4060056347351743577501971309, 10.269194303243351551435101529541, 10.701663429364387670645541940062, 11.75308532502535232447219405473, 11.95552354706698247156467878262, 13.07405733415550532301281340127, 13.69423025443139910089314087191, 14.42228222464994375262582725324, 15.68907335075253048104826685718, 16.197949711726464759294907955946, 16.83877241288302586356552994879, 17.65043855358243058787999511417, 18.696544435015553254966005888837, 18.94720098700154072683353215672, 19.69577349852606197222483734027

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