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Properties

Label 1-2001-2001.1133-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.0123 + 0.999i$
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  + (0.470 + 0.882i)2-s + (−0.557 + 0.830i)4-s + (0.794 + 0.607i)5-s + (0.986 − 0.162i)7-s + (−0.994 − 0.101i)8-s + (−0.162 + 0.986i)10-s + (0.0407 − 0.999i)11-s + (0.818 + 0.574i)13-s + (0.607 + 0.794i)14-s + (−0.377 − 0.925i)16-s + (0.989 − 0.142i)17-s + (−0.830 − 0.557i)19-s + (−0.947 + 0.320i)20-s + (0.900 − 0.433i)22-s + (0.262 + 0.965i)25-s + (−0.122 + 0.992i)26-s + ⋯
L(s)  = 1  + (0.470 + 0.882i)2-s + (−0.557 + 0.830i)4-s + (0.794 + 0.607i)5-s + (0.986 − 0.162i)7-s + (−0.994 − 0.101i)8-s + (−0.162 + 0.986i)10-s + (0.0407 − 0.999i)11-s + (0.818 + 0.574i)13-s + (0.607 + 0.794i)14-s + (−0.377 − 0.925i)16-s + (0.989 − 0.142i)17-s + (−0.830 − 0.557i)19-s + (−0.947 + 0.320i)20-s + (0.900 − 0.433i)22-s + (0.262 + 0.965i)25-s + (−0.122 + 0.992i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0123 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0123 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.0123 + 0.999i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.0123 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.949374001 + 1.973601578i\)
\(L(\frac12)\) \(\approx\) \(1.949374001 + 1.973601578i\)
\(L(1)\) \(\approx\) \(1.425157520 + 0.8990266153i\)
\(L(1)\) \(\approx\) \(1.425157520 + 0.8990266153i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.470 + 0.882i)T \)
5 \( 1 + (0.794 + 0.607i)T \)
7 \( 1 + (0.986 - 0.162i)T \)
11 \( 1 + (0.0407 - 0.999i)T \)
13 \( 1 + (0.818 + 0.574i)T \)
17 \( 1 + (0.989 - 0.142i)T \)
19 \( 1 + (-0.830 - 0.557i)T \)
31 \( 1 + (0.940 - 0.339i)T \)
37 \( 1 + (-0.242 + 0.970i)T \)
41 \( 1 + (-0.909 - 0.415i)T \)
43 \( 1 + (-0.940 - 0.339i)T \)
47 \( 1 + (0.781 - 0.623i)T \)
53 \( 1 + (-0.488 + 0.872i)T \)
59 \( 1 + (0.959 - 0.281i)T \)
61 \( 1 + (0.852 + 0.523i)T \)
67 \( 1 + (0.999 - 0.0407i)T \)
71 \( 1 + (0.182 - 0.983i)T \)
73 \( 1 + (0.0815 + 0.996i)T \)
79 \( 1 + (0.925 + 0.377i)T \)
83 \( 1 + (-0.0203 + 0.999i)T \)
89 \( 1 + (-0.699 - 0.714i)T \)
97 \( 1 + (-0.639 - 0.768i)T \)
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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

−20.07664822569350467361956811996, −19.08007668857006750253711155199, −18.32049025826068072232913233098, −17.67361881030442828034288352580, −17.19495470256854293918500965894, −16.055550018383653344767201850094, −15.059762594752493642125279645189, −14.52396138540820933367453346692, −13.825359181227534082964403098779, −13.00362405904478168014248041843, −12.43413380724146574048173413453, −11.79942494201758873267013096872, −10.8146461791474003640678357301, −10.18512711213732703361876183036, −9.57115605708493298119478274112, −8.544549122166126057704435207440, −8.08867914606417045497882308047, −6.63665893325312239802892949723, −5.73177241143062704733520584524, −5.15332438708177500208769771282, −4.442544825567574103480736709211, −3.557336433801600069873514301791, −2.36937118810228648510252920127, −1.69585531372996367872986741658, −1.01870994239226233303190573084, 1.07481544759701777769073134932, 2.26621560495995142219428026366, 3.25898392803661130125923763550, 4.04488357643217459032752123929, 5.07680161718038830731542935168, 5.69039025406586124151769743675, 6.50466265963883256595352664137, 7.06119847393818834559005267528, 8.25359700017254056741844916631, 8.53059854458645391136422019418, 9.586126359049820851447102071375, 10.54308581313969775609398861615, 11.33752380661456793821642853901, 11.99933361007018765807603984457, 13.21214433229690286544492790466, 13.797465301684438026051683457312, 14.13009373082822421467438384642, 14.98542160779886957050332798183, 15.58080314296463633133047201968, 16.73459045021642636358760715623, 17.00196110266830578243214604756, 17.819085398475432393256459899785, 18.64433610662076023150309609444, 18.93255750419214166388134924256, 20.51942529147736080963397364621

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