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Properties

Label 1-2001-2001.449-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.450 + 0.892i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.450 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.450 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.450 + 0.892i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.450 + 0.892i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3206972029 - 0.5209314836i\)
\(L(\frac12)\) \(\approx\) \(-0.3206972029 - 0.5209314836i\)
\(L(1)\) \(\approx\) \(0.5990204743 - 0.5948705271i\)
\(L(1)\) \(\approx\) \(0.5990204743 - 0.5948705271i\)

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

−20.61640816905939986617273999736, −19.34294639466256961996493412630, −18.86690853134680195507168972471, −18.09589047731947678862964419699, −17.73475946291039618336104944869, −16.56116177660849966225754845777, −15.8746076651694399584007367521, −15.417142804051429102326874937601, −14.72684570636330933979337558853, −14.05614735624328333078840135402, −13.28981029884709534485319882292, −12.41001051857849563373551755132, −11.609371831063534843929000185379, −10.90407990849598525865255103777, −10.01504485511216856260360799608, −8.79827327346373103325039483073, −8.224688152388965537685005952510, −7.90830027168057178681396717581, −6.65591481336292781966959387673, −6.14001503075587285099203150834, −5.36103591057577263107607665466, −4.44370725245445459390627346601, −3.46761830707426161960737607267, −2.94919424705586432136246673350, −1.40703728173538437645768909776, 0.21256037505331039355004718028, 1.27547832725129799071932049855, 2.03441792625657739869337313791, 3.25231940558921217986215687386, 4.09736574760934899241061948345, 4.69547224183483229116249638149, 5.18959663108173354869691686492, 6.6655667786482812597647739791, 7.47366845867360285741807300559, 8.42044980752182189488584882175, 9.0012731402501341168305905294, 9.88455786754429842352763576783, 10.6988412537222590772323312138, 11.49939898414627825353097606153, 11.83071393542355187312620221023, 12.83624418931357796250387728028, 13.5156102381502805906476698861, 13.962465961779446816616178342137, 15.14232372006241459303819269932, 15.63036604124083465297272154957, 16.67185122397904368326281777566, 17.49841322435598234623026892666, 18.01524693913873064238365678799, 18.92190210717688557772913474815, 19.68685222321086781149395116933

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