Properties

Label 1-2001-2001.446-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.760 - 0.649i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.760 - 0.649i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (446, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.760 - 0.649i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4366089060 - 0.1612077392i\)
\(L(\frac12)\) \(\approx\) \(0.4366089060 - 0.1612077392i\)
\(L(1)\) \(\approx\) \(0.5767177781 + 0.1899847436i\)
\(L(1)\) \(\approx\) \(0.5767177781 + 0.1899847436i\)

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

−20.006373060270480487204356209363, −19.48538871106568822715438663127, −18.80614833447931304215743524780, −17.855987779287423705928580382285, −17.40748610131107131946672860701, −16.43108155028539651392892818868, −15.86249331409675151630685977865, −14.56862362955997584136818954916, −14.29950416266944230608615281139, −13.19718831041213426689966595317, −12.48931250078271750586353670663, −11.87019845859445740867408815605, −11.12112478983466554387224162300, −10.41003513490062629702698440858, −9.7525078152659948370370398142, −9.05347393462561109665757459747, −7.990870367154042473027306339245, −7.24636431445663796594776639905, −6.79682007562852696559571960418, −5.31284192702015500545972647362, −4.236086743183731710789926052050, −3.806484694373605634293316733526, −2.96880462686636662342589162916, −2.06157250488764351220179445069, −0.82730034246669516263761970785, 0.25148478351596567882187052898, 1.4583081984329662701877843818, 2.801183867720110180246013924451, 3.94641690008916117694383859711, 4.61845508930082358832753393184, 5.46488997309926105635005393971, 6.33302590704143282862941175915, 7.005560978053061126401448299947, 7.83727348389031431968551920039, 8.71101353696041313838427786074, 9.18106956166597162076254802168, 9.75577852138853442335079217357, 11.09146474325765263143362557080, 11.73053057224306251892179548294, 12.63768766297350642694220025703, 13.25795521978416099931000523030, 14.219706048526428986489563205454, 15.03229794791440972313826436771, 15.528582905764067812523331349077, 16.23587498751661308217209248776, 16.82957294924703247523262442707, 17.48833768413442725888699747397, 18.44289846007128070459387500021, 19.19207647399298862369114993167, 19.59278278922818682644112059864

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