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Properties

Label 1-2001-2001.302-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.425 - 0.905i$
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  + (0.281 + 0.959i)2-s + (−0.841 + 0.540i)4-s + (−0.142 − 0.989i)5-s + (0.415 − 0.909i)7-s + (−0.755 − 0.654i)8-s + (0.909 − 0.415i)10-s + (−0.281 + 0.959i)11-s + (−0.415 − 0.909i)13-s + (0.989 + 0.142i)14-s + (0.415 − 0.909i)16-s + (0.540 − 0.841i)17-s + (0.540 + 0.841i)19-s + (0.654 + 0.755i)20-s − 22-s + (−0.959 + 0.281i)25-s + (0.755 − 0.654i)26-s + ⋯
L(s)  = 1  + (0.281 + 0.959i)2-s + (−0.841 + 0.540i)4-s + (−0.142 − 0.989i)5-s + (0.415 − 0.909i)7-s + (−0.755 − 0.654i)8-s + (0.909 − 0.415i)10-s + (−0.281 + 0.959i)11-s + (−0.415 − 0.909i)13-s + (0.989 + 0.142i)14-s + (0.415 − 0.909i)16-s + (0.540 − 0.841i)17-s + (0.540 + 0.841i)19-s + (0.654 + 0.755i)20-s − 22-s + (−0.959 + 0.281i)25-s + (0.755 − 0.654i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.425 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.425 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.425 - 0.905i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (302, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.425 - 0.905i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9825315982 - 0.6240482829i\)
\(L(\frac12)\) \(\approx\) \(0.9825315982 - 0.6240482829i\)
\(L(1)\) \(\approx\) \(1.009907820 + 0.1152117959i\)
\(L(1)\) \(\approx\) \(1.009907820 + 0.1152117959i\)

Euler product

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

−19.860810882459833449693945932647, −19.35861921351241255832385067942, −18.72146573250615065363812278189, −18.2029325373660884807606043565, −17.53195405798416121610496222197, −16.38665534679699283633758029794, −15.49540839526603257971282157583, −14.59708378253251055434892288135, −14.37426241686627147237830923864, −13.44439044918028198829958181931, −12.60645943840177730258752166617, −11.74894295619070731269570686981, −11.27934029441493009504711855895, −10.69331002585092877185699098721, −9.75412313836355437029346533235, −9.02634791231961633933944335013, −8.26251574289541218506941013693, −7.29387432294380269345616338869, −6.1347826010311369239380376223, −5.61353154659363502337747558585, −4.622061569816478901178109661569, −3.69688442733066546814020361698, −2.85810757073369272980649771497, −2.31264416731072858879953587673, −1.25368608532828587603589448917, 0.391843618324879613967283668232, 1.41873906284357305168866565955, 2.89456896228836957919052313625, 3.99557373831184538370915988538, 4.583809859589823015936417643, 5.28774016176951045936303446560, 5.95696579423864364538736736597, 7.38008985892520136448728406262, 7.5709019057216567250506502265, 8.21532778620466275448748209529, 9.46691629291085092907440627240, 9.738902555946959315900173096936, 10.90058878332183598547816017672, 12.08164400949915778656215355720, 12.58473645416768111601506225158, 13.24647179997438782398030420861, 14.08394060897828649048060114685, 14.67084692067645032219683591201, 15.55533421154079447158550063716, 16.185041296793873559775450102812, 16.86824123022196646002243176723, 17.45507376009907937268170345317, 18.038240174537719401362358433620, 18.94014958991935248258221937235, 20.29849656452430349624769981379

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