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Properties

Label 1-2001-2001.1520-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.790 - 0.612i$
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  + (0.909 − 0.415i)2-s + (0.654 − 0.755i)4-s + (0.841 + 0.540i)5-s + (−0.142 − 0.989i)7-s + (0.281 − 0.959i)8-s + (0.989 + 0.142i)10-s + (−0.909 − 0.415i)11-s + (0.142 − 0.989i)13-s + (−0.540 − 0.841i)14-s + (−0.142 − 0.989i)16-s + (−0.755 + 0.654i)17-s + (−0.755 − 0.654i)19-s + (0.959 − 0.281i)20-s − 22-s + (0.415 + 0.909i)25-s + (−0.281 − 0.959i)26-s + ⋯
L(s)  = 1  + (0.909 − 0.415i)2-s + (0.654 − 0.755i)4-s + (0.841 + 0.540i)5-s + (−0.142 − 0.989i)7-s + (0.281 − 0.959i)8-s + (0.989 + 0.142i)10-s + (−0.909 − 0.415i)11-s + (0.142 − 0.989i)13-s + (−0.540 − 0.841i)14-s + (−0.142 − 0.989i)16-s + (−0.755 + 0.654i)17-s + (−0.755 − 0.654i)19-s + (0.959 − 0.281i)20-s − 22-s + (0.415 + 0.909i)25-s + (−0.281 − 0.959i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.790 - 0.612i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1520, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.790 - 0.612i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8072875990 - 2.360196876i\)
\(L(\frac12)\) \(\approx\) \(0.8072875990 - 2.360196876i\)
\(L(1)\) \(\approx\) \(1.508024377 - 0.8944322490i\)
\(L(1)\) \(\approx\) \(1.508024377 - 0.8944322490i\)

Euler product

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

−20.59201295811262444796673655292, −19.68025750657165021516320494307, −18.62273547091847089521091633473, −17.968562639025767143516413170905, −17.217058683483247797887954319062, −16.34562724638743933673425519944, −15.90551171352737422632559579030, −15.07643536452385752174256426349, −14.38285816897635432708279508825, −13.48199757888817162937383275835, −13.1081220976200706632667146573, −12.20282838179679517457550642378, −11.77511676965264918291391700139, −10.64393564758719817694772614827, −9.805449993393294496722726328623, −8.7307790721768470642453696850, −8.43547799724724938445639130121, −7.1159859131735800349217146388, −6.46697322952732440157756840110, −5.694916008411602885736407145494, −4.99356728573818880364457123339, −4.429744740087324456992346065393, −3.16838382747710092345326871669, −2.253127756689675491488054545498, −1.79084236787413810125465586730, 0.52869904349917344118117273023, 1.7764608559380127742553750513, 2.60218401642131610415212576065, 3.320272641555957876568937145027, 4.21853725936418360854399705243, 5.12256511346873986324064410263, 5.91853969319844389076521301146, 6.55942480552744468649810138950, 7.34305436796206342104792101893, 8.32003177280436332199594397388, 9.57110295041310010740194984239, 10.309980123427111866751682595774, 10.81184609182843427879112807928, 11.23660146688620395524607437164, 12.77858405738770251902648666505, 13.00267270452432717038981723145, 13.70820705038701045313144898345, 14.292251602153335926918345618733, 15.29809045669427617413780747177, 15.63260913702213562109819619421, 16.845619974681909488250865005583, 17.42885908908114049679153170857, 18.312908093921980634735348564360, 19.06575213266631327011245133902, 19.86362367872012806037605170817

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