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Properties

Label 1-2001-2001.332-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.672 - 0.739i$
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  + (−0.862 + 0.505i)2-s + (0.488 − 0.872i)4-s + (0.182 − 0.983i)5-s + (−0.339 + 0.940i)7-s + (0.0203 + 0.999i)8-s + (0.339 + 0.940i)10-s + (0.301 − 0.953i)11-s + (−0.992 + 0.122i)13-s + (−0.182 − 0.983i)14-s + (−0.523 − 0.852i)16-s + (0.959 − 0.281i)17-s + (0.488 − 0.872i)19-s + (−0.768 − 0.639i)20-s + (0.222 + 0.974i)22-s + (−0.933 − 0.359i)25-s + (0.794 − 0.607i)26-s + ⋯
L(s)  = 1  + (−0.862 + 0.505i)2-s + (0.488 − 0.872i)4-s + (0.182 − 0.983i)5-s + (−0.339 + 0.940i)7-s + (0.0203 + 0.999i)8-s + (0.339 + 0.940i)10-s + (0.301 − 0.953i)11-s + (−0.992 + 0.122i)13-s + (−0.182 − 0.983i)14-s + (−0.523 − 0.852i)16-s + (0.959 − 0.281i)17-s + (0.488 − 0.872i)19-s + (−0.768 − 0.639i)20-s + (0.222 + 0.974i)22-s + (−0.933 − 0.359i)25-s + (0.794 − 0.607i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.672 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.672 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.672 - 0.739i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (332, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.672 - 0.739i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1940137592 - 0.4386823534i\)
\(L(\frac12)\) \(\approx\) \(0.1940137592 - 0.4386823534i\)
\(L(1)\) \(\approx\) \(0.6279951397 - 0.05395665274i\)
\(L(1)\) \(\approx\) \(0.6279951397 - 0.05395665274i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.862 + 0.505i)T \)
5 \( 1 + (0.182 - 0.983i)T \)
7 \( 1 + (-0.339 + 0.940i)T \)
11 \( 1 + (0.301 - 0.953i)T \)
13 \( 1 + (-0.992 + 0.122i)T \)
17 \( 1 + (0.959 - 0.281i)T \)
19 \( 1 + (0.488 - 0.872i)T \)
31 \( 1 + (-0.970 + 0.242i)T \)
37 \( 1 + (0.262 - 0.965i)T \)
41 \( 1 + (-0.654 - 0.755i)T \)
43 \( 1 + (0.970 + 0.242i)T \)
47 \( 1 + (-0.900 - 0.433i)T \)
53 \( 1 + (0.101 + 0.994i)T \)
59 \( 1 + (-0.841 + 0.540i)T \)
61 \( 1 + (-0.979 - 0.202i)T \)
67 \( 1 + (0.301 + 0.953i)T \)
71 \( 1 + (-0.557 + 0.830i)T \)
73 \( 1 + (0.818 - 0.574i)T \)
79 \( 1 + (-0.523 + 0.852i)T \)
83 \( 1 + (-0.591 - 0.806i)T \)
89 \( 1 + (0.452 - 0.891i)T \)
97 \( 1 + (0.882 - 0.470i)T \)
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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

−20.02202816279026380630743689001, −19.53904151663679521888318803404, −18.7205721930611982140280313428, −18.158713581653781131163050819682, −17.28869851532361498069223561572, −16.90153958704230937222241312578, −16.0939680966382449581698409407, −14.99718227118867991284996765608, −14.501532873436764830825289712200, −13.56529889378747172091790591574, −12.61289780213655893278425663421, −12.04618524395679640339442497753, −11.14513924522957153784321687838, −10.379209849782611560409733563198, −9.81622063529626447421210456873, −9.50103505158065150628991417009, −7.95437421904975702657875305068, −7.545025286582096228034418615516, −6.884273194612870181394560120932, −6.10432645037359278346995433307, −4.73482279577133809627284632859, −3.65850880078569393416656969972, −3.16079307748953558998976261423, −2.09405106874907132122957786841, −1.297753929054165982044413127113, 0.23106650963057340762492550340, 1.267742508746894300864764236, 2.276293402664829847381705330431, 3.18179683981386759656212942381, 4.631793967144703959642029273106, 5.54538672694016237625985936610, 5.78120319075008154462804987483, 6.95947147209607735373192313025, 7.72853234983447807948329183390, 8.6043327272854710292155413440, 9.22025521440420284937705913222, 9.56643949810335896717920703196, 10.6097831739817718602856982963, 11.62945409773387430848667138287, 12.12745685699763577051629196290, 13.023806796477775553837373799507, 14.035367055720557195300291795689, 14.64026255972058263034859226293, 15.64740670637870682054038989400, 16.116302518531871780798830432741, 16.79169695994576744669807659854, 17.35040534404216287201078789213, 18.27056878713320380819268302794, 18.86128848852594841536214374760, 19.68012595495745263050833570498

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