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Properties

Label 1-2001-2001.476-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.778 + 0.627i$
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  + (−0.989 + 0.142i)2-s + (0.959 − 0.281i)4-s + (−0.654 + 0.755i)5-s + (0.841 − 0.540i)7-s + (−0.909 + 0.415i)8-s + (0.540 − 0.841i)10-s + (0.989 + 0.142i)11-s + (−0.841 − 0.540i)13-s + (−0.755 + 0.654i)14-s + (0.841 − 0.540i)16-s + (−0.281 + 0.959i)17-s + (−0.281 − 0.959i)19-s + (−0.415 + 0.909i)20-s − 22-s + (−0.142 − 0.989i)25-s + (0.909 + 0.415i)26-s + ⋯
L(s)  = 1  + (−0.989 + 0.142i)2-s + (0.959 − 0.281i)4-s + (−0.654 + 0.755i)5-s + (0.841 − 0.540i)7-s + (−0.909 + 0.415i)8-s + (0.540 − 0.841i)10-s + (0.989 + 0.142i)11-s + (−0.841 − 0.540i)13-s + (−0.755 + 0.654i)14-s + (0.841 − 0.540i)16-s + (−0.281 + 0.959i)17-s + (−0.281 − 0.959i)19-s + (−0.415 + 0.909i)20-s − 22-s + (−0.142 − 0.989i)25-s + (0.909 + 0.415i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.778 + 0.627i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (476, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.778 + 0.627i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1410813324 + 0.3998986165i\)
\(L(\frac12)\) \(\approx\) \(0.1410813324 + 0.3998986165i\)
\(L(1)\) \(\approx\) \(0.5841777236 + 0.1163300676i\)
\(L(1)\) \(\approx\) \(0.5841777236 + 0.1163300676i\)

Euler product

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

−19.62532378175650885314373692432, −18.934693417347603014873503865744, −18.31033868514107232746558298102, −17.40754162427386523609311751912, −16.80521725859209656079897624254, −16.29764253537254321025808188910, −15.39439845213594725933622507136, −14.7539905596657445284948027936, −13.96850798395021304176395200258, −12.58510247837033715199531112456, −12.05873630292575246111586651145, −11.57375892388130769056003038035, −10.8867751185001563370855180009, −9.71704832120124563179087934222, −9.0756860253344927232904978675, −8.59329940088587806245312697102, −7.70001904118033642156531369935, −7.15894199279638116667305724496, −6.085267368987726976002279494211, −5.11219549978524871618780379672, −4.24527065342686926421458455402, −3.30499912905606312917456678052, −2.038214981154353487203754319478, −1.50691758450369845655085447344, −0.21442212843128701525757886768, 1.14254936133011586287729696844, 2.06087602708348001296637446196, 3.04477999578570456950935255656, 4.00670419226904000897743919986, 4.934459588144651074087992768206, 6.142100238542150741065617177685, 6.93548096689109880949002206649, 7.43735134075358793860489867658, 8.1888949310718203990841766588, 8.92562822865123878259030444330, 9.85422948498799222002170785148, 10.78691008891697196285787099277, 10.98513244676471769252953091157, 11.91427749077352099894420424201, 12.573060471317523313654934014455, 13.987742259806289500103940395930, 14.64151938367977760089533662915, 15.15304750769650460575920251095, 15.80354157121971415606834241502, 16.973024642089035836569468100362, 17.32638046337106061257779123049, 17.92043691228643714903778938651, 18.8265395075760146839545460564, 19.58034215345278208658788934463, 19.880794433489366867770982803581

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