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Properties

Label 1-2001-2001.1007-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.665 + 0.746i$
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  + (0.320 + 0.947i)2-s + (−0.794 + 0.607i)4-s + (−0.818 + 0.574i)5-s + (−0.591 − 0.806i)7-s + (−0.830 − 0.557i)8-s + (−0.806 − 0.591i)10-s + (−0.852 − 0.523i)11-s + (−0.917 − 0.396i)13-s + (0.574 − 0.818i)14-s + (0.262 − 0.965i)16-s + (0.909 + 0.415i)17-s + (−0.607 − 0.794i)19-s + (0.301 − 0.953i)20-s + (0.222 − 0.974i)22-s + (0.339 − 0.940i)25-s + (0.0815 − 0.996i)26-s + ⋯
L(s)  = 1  + (0.320 + 0.947i)2-s + (−0.794 + 0.607i)4-s + (−0.818 + 0.574i)5-s + (−0.591 − 0.806i)7-s + (−0.830 − 0.557i)8-s + (−0.806 − 0.591i)10-s + (−0.852 − 0.523i)11-s + (−0.917 − 0.396i)13-s + (0.574 − 0.818i)14-s + (0.262 − 0.965i)16-s + (0.909 + 0.415i)17-s + (−0.607 − 0.794i)19-s + (0.301 − 0.953i)20-s + (0.222 − 0.974i)22-s + (0.339 − 0.940i)25-s + (0.0815 − 0.996i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.665 + 0.746i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.665 + 0.746i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2556824814 + 0.5702569839i\)
\(L(\frac12)\) \(\approx\) \(0.2556824814 + 0.5702569839i\)
\(L(1)\) \(\approx\) \(0.6478256848 + 0.3447382994i\)
\(L(1)\) \(\approx\) \(0.6478256848 + 0.3447382994i\)

Euler product

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

−19.701735943745239408024822079412, −18.971486879893329795069939679016, −18.68961233829208893762305001448, −17.67090273428932409878513059734, −16.73780062991193057001699151176, −15.93570459457670116384897976166, −15.208495622470907202428051695324, −14.58024826427693890494543089631, −13.63955060241670516187413024716, −12.63457694035662867601801780148, −12.33264735943135544283801840827, −11.89062417241712430427425489080, −10.83148429747728548048673775623, −9.99672353301750336889879511012, −9.42573879404804901404701970417, −8.55746146526288160368551586652, −7.841887800898581113880549464891, −6.79199858348386757627830320300, −5.55158874340343651202496675511, −5.08630411203598600359368674287, −4.21904147382344043928582390004, −3.34195803089226435735845738425, −2.55024210183331646560093180911, −1.68656283510574898870548474993, −0.303918612634849285231751570366, 0.67559269145286926957659143945, 2.7692677038035587208746677101, 3.23855936344740804076543531490, 4.19452200108586305740101788080, 4.88673194137331984028339553579, 5.919012647233151442366560045212, 6.687163308779562188346520188737, 7.420632186595829006473751173235, 7.92203070583395489993632050722, 8.67481582929010951889945351867, 9.96877027523192296020305448355, 10.330634667588725489144210989545, 11.45394171037736436950679135417, 12.27442622516302009608530881867, 13.11667379954108930048288823809, 13.58644660821225194485160927720, 14.63147301916659337336041043537, 15.058346405417513434699114624144, 15.82828554939258821055494249685, 16.473713934484346111782740861967, 17.134241742565156217455243687377, 17.86441682966216056016018309675, 18.9438263014832681432795077596, 19.160084677250058213908112631841, 20.16889877457394725224269363242

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