Properties

Label 1-2001-2001.2-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.563 - 0.825i$
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  + (−0.202 − 0.979i)2-s + (−0.917 + 0.396i)4-s + (−0.714 + 0.699i)5-s + (0.557 + 0.830i)7-s + (0.574 + 0.818i)8-s + (0.830 + 0.557i)10-s + (0.242 + 0.970i)11-s + (0.862 − 0.505i)13-s + (0.699 − 0.714i)14-s + (0.685 − 0.728i)16-s + (0.755 − 0.654i)17-s + (−0.396 − 0.917i)19-s + (0.377 − 0.925i)20-s + (0.900 − 0.433i)22-s + (0.0203 − 0.999i)25-s + (−0.670 − 0.742i)26-s + ⋯
L(s)  = 1  + (−0.202 − 0.979i)2-s + (−0.917 + 0.396i)4-s + (−0.714 + 0.699i)5-s + (0.557 + 0.830i)7-s + (0.574 + 0.818i)8-s + (0.830 + 0.557i)10-s + (0.242 + 0.970i)11-s + (0.862 − 0.505i)13-s + (0.699 − 0.714i)14-s + (0.685 − 0.728i)16-s + (0.755 − 0.654i)17-s + (−0.396 − 0.917i)19-s + (0.377 − 0.925i)20-s + (0.900 − 0.433i)22-s + (0.0203 − 0.999i)25-s + (−0.670 − 0.742i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.563 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.563 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.563 - 0.825i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.563 - 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.114730639 - 0.5887414167i\)
\(L(\frac12)\) \(\approx\) \(1.114730639 - 0.5887414167i\)
\(L(1)\) \(\approx\) \(0.8727276814 - 0.2589975435i\)
\(L(1)\) \(\approx\) \(0.8727276814 - 0.2589975435i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.202 - 0.979i)T \)
5 \( 1 + (-0.714 + 0.699i)T \)
7 \( 1 + (0.557 + 0.830i)T \)
11 \( 1 + (0.242 + 0.970i)T \)
13 \( 1 + (0.862 - 0.505i)T \)
17 \( 1 + (0.755 - 0.654i)T \)
19 \( 1 + (-0.396 - 0.917i)T \)
31 \( 1 + (-0.872 - 0.488i)T \)
37 \( 1 + (0.994 + 0.101i)T \)
41 \( 1 + (0.540 - 0.841i)T \)
43 \( 1 + (0.872 - 0.488i)T \)
47 \( 1 + (0.781 - 0.623i)T \)
53 \( 1 + (-0.996 - 0.0815i)T \)
59 \( 1 + (0.142 - 0.989i)T \)
61 \( 1 + (-0.162 - 0.986i)T \)
67 \( 1 + (-0.970 - 0.242i)T \)
71 \( 1 + (-0.452 + 0.891i)T \)
73 \( 1 + (-0.470 + 0.882i)T \)
79 \( 1 + (0.728 - 0.685i)T \)
83 \( 1 + (0.992 - 0.122i)T \)
89 \( 1 + (0.998 - 0.0611i)T \)
97 \( 1 + (0.852 - 0.523i)T \)
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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

−19.84426716654298378794688072811, −19.245636754800971365984845648877, −18.59031691989090649815398978718, −17.732601965275285968503504494909, −16.842093844264898497268010617067, −16.43062727535228274024848773442, −16.038004721352842649303146148562, −14.830552704762069940697233676137, −14.43243551538542237744526489707, −13.575557821162995923458597271133, −12.93699293007812034215102672578, −11.99858177997478007813868777920, −11.01067641376871792359409120067, −10.47416313609332722327245086147, −9.24527564419374916338659597709, −8.70683670685716642309111851962, −7.82550042481593820416875600193, −7.62544953001992927059349909389, −6.2981345057066498799871172073, −5.81008449602600014971845066566, −4.71870053420024302425887298066, −4.030584735683480264658459624129, −3.49973966524337798687121096729, −1.425134996976448647881564969118, −0.94146339736478801579846097142, 0.64853709650606050719464348222, 1.91257626016416933920139930370, 2.61574591522265853885514658585, 3.45741920444317692525026049812, 4.29149437969964785739563565857, 5.08421509954793108963254771580, 6.06759555689761974790040043252, 7.32890970309601657389189013016, 7.83780766923029163316480279161, 8.777284996723844250141257583553, 9.39221434110742761761312701725, 10.35690469689436442305292471656, 11.090378250408057556904177335732, 11.568557858829622153647014357598, 12.333569280889335786842764527469, 12.91993622146063286020469106820, 14.01237917216050722981443457540, 14.67038983607006065392253516222, 15.34940312459006904931456116130, 16.104963345772756975613518288062, 17.33192819855821684190194782448, 17.86095601790037534659116825843, 18.611532095357891147341800496245, 18.92539946776672326057570120832, 19.94520043679775596031602299068

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