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Properties

Label 1-2001-2001.1118-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.853 - 0.520i$
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  + (−0.947 − 0.320i)2-s + (0.794 + 0.607i)4-s + (−0.818 − 0.574i)5-s + (0.591 − 0.806i)7-s + (−0.557 − 0.830i)8-s + (0.591 + 0.806i)10-s + (−0.523 − 0.852i)11-s + (0.917 − 0.396i)13-s + (−0.818 + 0.574i)14-s + (0.262 + 0.965i)16-s + (0.415 + 0.909i)17-s + (−0.794 − 0.607i)19-s + (−0.301 − 0.953i)20-s + (0.222 + 0.974i)22-s + (0.339 + 0.940i)25-s + (−0.996 + 0.0815i)26-s + ⋯
L(s)  = 1  + (−0.947 − 0.320i)2-s + (0.794 + 0.607i)4-s + (−0.818 − 0.574i)5-s + (0.591 − 0.806i)7-s + (−0.557 − 0.830i)8-s + (0.591 + 0.806i)10-s + (−0.523 − 0.852i)11-s + (0.917 − 0.396i)13-s + (−0.818 + 0.574i)14-s + (0.262 + 0.965i)16-s + (0.415 + 0.909i)17-s + (−0.794 − 0.607i)19-s + (−0.301 − 0.953i)20-s + (0.222 + 0.974i)22-s + (0.339 + 0.940i)25-s + (−0.996 + 0.0815i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.853 - 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.853 - 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.853 - 0.520i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.853 - 0.520i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2101068252 - 0.7484874837i\)
\(L(\frac12)\) \(\approx\) \(0.2101068252 - 0.7484874837i\)
\(L(1)\) \(\approx\) \(0.5799878386 - 0.3117202366i\)
\(L(1)\) \(\approx\) \(0.5799878386 - 0.3117202366i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.947 - 0.320i)T \)
5 \( 1 + (-0.818 - 0.574i)T \)
7 \( 1 + (0.591 - 0.806i)T \)
11 \( 1 + (-0.523 - 0.852i)T \)
13 \( 1 + (0.917 - 0.396i)T \)
17 \( 1 + (0.415 + 0.909i)T \)
19 \( 1 + (-0.794 - 0.607i)T \)
31 \( 1 + (0.685 - 0.728i)T \)
37 \( 1 + (-0.986 + 0.162i)T \)
41 \( 1 + (0.959 - 0.281i)T \)
43 \( 1 + (-0.685 - 0.728i)T \)
47 \( 1 + (0.900 + 0.433i)T \)
53 \( 1 + (0.182 - 0.983i)T \)
59 \( 1 + (0.654 - 0.755i)T \)
61 \( 1 + (0.933 - 0.359i)T \)
67 \( 1 + (0.523 - 0.852i)T \)
71 \( 1 + (0.992 - 0.122i)T \)
73 \( 1 + (-0.452 - 0.891i)T \)
79 \( 1 + (-0.262 + 0.965i)T \)
83 \( 1 + (0.488 - 0.872i)T \)
89 \( 1 + (-0.862 + 0.505i)T \)
97 \( 1 + (0.0611 + 0.998i)T \)
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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

−20.15054373067269565538580755456, −19.21937505599344047326362866482, −18.69663503172079300493230388694, −18.17177083908086857949225208773, −17.59237852019392011519377557563, −16.518861063015416372168732210719, −15.80372503571096682278620024014, −15.38202962080607247166109475263, −14.6221715962004924692304505485, −14.03147628299519883378282256304, −12.598783500661072665711810402799, −11.882463716724525310431112650654, −11.318979138114319615094930718621, −10.54338275861905193334897499439, −9.86638491944652967472564021566, −8.77612981148690577845869260648, −8.34134008059406997763684766297, −7.512336851902915923285148225067, −6.89106844972196752698218389959, −5.99563654604250829487323907728, −5.125027258556357356766883927644, −4.14236992989029156316374211530, −2.90358013055964106328997654927, −2.202669904258447365164932648590, −1.16139573270876319693327327514, 0.45316914592349150594447403952, 1.13071589306355975050439139327, 2.22555795718554502223590179390, 3.525452315126265576987825690572, 3.88433699351220687259304490227, 5.05770828898465159551314691775, 6.128724954328825553248643355131, 7.052285531328597604005284491356, 8.051704748775461873469565771981, 8.226422822933750233444910012640, 8.950429336760426649616784866349, 10.13350604981016589175553129418, 10.86995725161071331911537270793, 11.17835039020439555585314262209, 12.10581755994188083107297023781, 12.95204089397138623137434337284, 13.52776101952344543412442966384, 14.73311890631652646266011402603, 15.62552135831705021521666637232, 16.02486884801904888037462761135, 17.06614421400563198068096214849, 17.223245529376590550197601723519, 18.304888865945557724178635189385, 19.04772792976511383719969261403, 19.521304539844091992237968983680

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