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Properties

Label 1-2001-2001.380-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.0797 - 0.996i$
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  + (−0.999 + 0.0203i)2-s + (0.999 − 0.0407i)4-s + (−0.523 + 0.852i)5-s + (−0.862 + 0.505i)7-s + (−0.998 + 0.0611i)8-s + (0.505 − 0.862i)10-s + (0.607 + 0.794i)11-s + (0.933 − 0.359i)13-s + (0.852 − 0.523i)14-s + (0.996 − 0.0815i)16-s + (−0.755 − 0.654i)17-s + (0.0407 + 0.999i)19-s + (−0.488 + 0.872i)20-s + (−0.623 − 0.781i)22-s + (−0.452 − 0.891i)25-s + (−0.925 + 0.377i)26-s + ⋯
L(s)  = 1  + (−0.999 + 0.0203i)2-s + (0.999 − 0.0407i)4-s + (−0.523 + 0.852i)5-s + (−0.862 + 0.505i)7-s + (−0.998 + 0.0611i)8-s + (0.505 − 0.862i)10-s + (0.607 + 0.794i)11-s + (0.933 − 0.359i)13-s + (0.852 − 0.523i)14-s + (0.996 − 0.0815i)16-s + (−0.755 − 0.654i)17-s + (0.0407 + 0.999i)19-s + (−0.488 + 0.872i)20-s + (−0.623 − 0.781i)22-s + (−0.452 − 0.891i)25-s + (−0.925 + 0.377i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0797 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0797 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.0797 - 0.996i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (380, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.0797 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1806010315 - 0.1667349814i\)
\(L(\frac12)\) \(\approx\) \(0.1806010315 - 0.1667349814i\)
\(L(1)\) \(\approx\) \(0.5092555181 + 0.09851061653i\)
\(L(1)\) \(\approx\) \(0.5092555181 + 0.09851061653i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.999 + 0.0203i)T \)
5 \( 1 + (-0.523 + 0.852i)T \)
7 \( 1 + (-0.862 + 0.505i)T \)
11 \( 1 + (0.607 + 0.794i)T \)
13 \( 1 + (0.933 - 0.359i)T \)
17 \( 1 + (-0.755 - 0.654i)T \)
19 \( 1 + (0.0407 + 0.999i)T \)
31 \( 1 + (-0.670 - 0.742i)T \)
37 \( 1 + (-0.699 - 0.714i)T \)
41 \( 1 + (-0.540 - 0.841i)T \)
43 \( 1 + (0.670 - 0.742i)T \)
47 \( 1 + (-0.974 + 0.222i)T \)
53 \( 1 + (0.301 + 0.953i)T \)
59 \( 1 + (0.142 + 0.989i)T \)
61 \( 1 + (-0.574 + 0.818i)T \)
67 \( 1 + (-0.794 - 0.607i)T \)
71 \( 1 + (-0.979 - 0.202i)T \)
73 \( 1 + (-0.965 + 0.262i)T \)
79 \( 1 + (0.0815 - 0.996i)T \)
83 \( 1 + (-0.947 + 0.320i)T \)
89 \( 1 + (0.162 + 0.986i)T \)
97 \( 1 + (0.994 + 0.101i)T \)
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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

−19.80557898351643783705982998242, −19.53528642117525662913890235482, −18.84372834352636061707299124778, −17.88062626043547244670278506046, −17.10567816222066481041056653266, −16.531588644443849947316003455852, −15.95369213944645201107925563360, −15.47032335559137704478227194692, −14.33139711159460764033312633483, −13.22467848286669063212104890315, −12.85625214583260046225052681530, −11.665269662591181463691476579, −11.282355488118166727569997650770, −10.43506318784763043604888981818, −9.50084757137570352950757682252, −8.75762275331016884367851113362, −8.49118434735113914636544912658, −7.3702883014530972793289264894, −6.57690454730587377801880433584, −6.046656391195584610919875491274, −4.75085215884870188850252313336, −3.69716002620198444666917231700, −3.15970401112213459674509082981, −1.70042606409302561392085610134, −0.946105706954885358015548379861, 0.14068090279020191991427849478, 1.62552682926588228424720823885, 2.510157787502218253305429571013, 3.35507342894063808040363283367, 4.08526414323324128844602591574, 5.71351058328851016906273192222, 6.286410657416123308613997450763, 7.11125541894906502408567697263, 7.595165902102154184729014510, 8.750440998413461830960330917610, 9.19142257572411990595305946122, 10.19504285489276296743962550621, 10.63182830900570146455930847222, 11.64679049669462383574472871499, 12.07215331656177624518028078403, 12.992412867000751263304444467207, 14.10570189022571603350160670788, 14.99749174923348472462978851495, 15.52126613655511182370712127926, 16.09998002502418016564190700707, 16.864104753114814140049376895073, 17.92624908359313208931553477879, 18.26702216282392658289062984480, 19.01992907065238045266667324562, 19.56804262874892281073769128846

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