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Properties

Label 1-2001-2001.1346-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.961 - 0.275i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.961 - 0.275i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1346, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.961 - 0.275i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.446519894 - 0.2028359003i\)
\(L(\frac12)\) \(\approx\) \(1.446519894 - 0.2028359003i\)
\(L(1)\) \(\approx\) \(0.9479550489 - 0.1175974141i\)
\(L(1)\) \(\approx\) \(0.9479550489 - 0.1175974141i\)

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

−19.97144891457763955633675247463, −19.176035974202793302522587557309, −18.23413033784092654643201006321, −17.89738499467605520189536549370, −17.04945000651616823530944722245, −16.637895384758375035432535899485, −15.75673259240846537412914952946, −14.81616192810012829247266312133, −14.226347533792800675961454795735, −13.712107895246330554000054451374, −12.588942516371146693154532962660, −11.6297921797929807692027202250, −10.775308919903020589529063565092, −10.10007862913591281215009609766, −9.723857794326062943493072866452, −8.874711225631323670146083565252, −7.73194909268664049082658140185, −7.2787305553210894074952257193, −6.48411188710005046362193127198, −5.76830631843227092772883381913, −4.93035036988621480656623055286, −3.5564456831301235597167423475, −2.78232796187051591788839075714, −1.54030095730719661372912517509, −0.94401317149519293157865304213, 0.922918104717191117216303859844, 1.7485595738738790274348310293, 2.478027382114207122512787906830, 3.47592875886885229845560561284, 4.46820093702334547671046350615, 5.7573703432909394293798803527, 6.18394543855617694391049921428, 7.15468309299598500222155418001, 8.20582398649741432258878041688, 8.97828956749257330271813737295, 9.34296986063852916713631403043, 9.954361930519040537969702084521, 11.12220830112818784909578521674, 11.68365150016377277422088316448, 12.41332095127237484624317929532, 13.0792530884093398311979772706, 14.03322240112619614357519795350, 14.82783546448751451367452757500, 15.87431482247226172072177593411, 16.513003118599317229245499316258, 17.0086436372707946541495461934, 17.850495742842327355217102242924, 18.42761431289621814559737723925, 19.23752915702449442582595978871, 19.677439903485314273808042726613

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