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Properties

Label 2-2001-2001.1034-c0-0-5
Degree $2$
Conductor $2001$
Sign $0.310 + 0.950i$
Analytic cond. $0.998629$
Root an. cond. $0.999314$
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  + (1.69 − 1.06i)2-s + (0.680 + 0.733i)3-s + (1.29 − 2.69i)4-s + (1.93 + 0.517i)6-s + (−0.446 − 3.96i)8-s + (−0.0747 + 0.997i)9-s + (2.86 − 0.882i)12-s + (−1.14 + 0.914i)13-s + (−3.10 − 3.88i)16-s + (0.933 + 1.76i)18-s + (−0.974 + 0.222i)23-s + (2.60 − 3.02i)24-s + (−0.900 − 0.433i)25-s + (−0.967 + 2.76i)26-s + (−0.781 + 0.623i)27-s + ⋯
L(s)  = 1  + (1.69 − 1.06i)2-s + (0.680 + 0.733i)3-s + (1.29 − 2.69i)4-s + (1.93 + 0.517i)6-s + (−0.446 − 3.96i)8-s + (−0.0747 + 0.997i)9-s + (2.86 − 0.882i)12-s + (−1.14 + 0.914i)13-s + (−3.10 − 3.88i)16-s + (0.933 + 1.76i)18-s + (−0.974 + 0.222i)23-s + (2.60 − 3.02i)24-s + (−0.900 − 0.433i)25-s + (−0.967 + 2.76i)26-s + (−0.781 + 0.623i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.310 + 0.950i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1034, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ 0.310 + 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.446708845\)
\(L(\frac12)\) \(\approx\) \(3.446708845\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.680 - 0.733i)T \)
23 \( 1 + (0.974 - 0.222i)T \)
29 \( 1 + (-0.563 - 0.826i)T \)
good2 \( 1 + (-1.69 + 1.06i)T + (0.433 - 0.900i)T^{2} \)
5 \( 1 + (0.900 + 0.433i)T^{2} \)
7 \( 1 + (-0.623 + 0.781i)T^{2} \)
11 \( 1 + (0.974 + 0.222i)T^{2} \)
13 \( 1 + (1.14 - 0.914i)T + (0.222 - 0.974i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (-0.781 + 0.623i)T^{2} \)
31 \( 1 + (-1.02 - 1.63i)T + (-0.433 + 0.900i)T^{2} \)
37 \( 1 + (-0.974 + 0.222i)T^{2} \)
41 \( 1 + (-0.565 + 0.565i)T - iT^{2} \)
43 \( 1 + (-0.433 - 0.900i)T^{2} \)
47 \( 1 + (-0.369 - 0.0416i)T + (0.974 + 0.222i)T^{2} \)
53 \( 1 + (-0.900 - 0.433i)T^{2} \)
59 \( 1 + 1.80iT - T^{2} \)
61 \( 1 + (0.781 + 0.623i)T^{2} \)
67 \( 1 + (-0.222 - 0.974i)T^{2} \)
71 \( 1 + (0.367 + 0.460i)T + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (0.806 - 1.28i)T + (-0.433 - 0.900i)T^{2} \)
79 \( 1 + (0.974 - 0.222i)T^{2} \)
83 \( 1 + (0.623 + 0.781i)T^{2} \)
89 \( 1 + (-0.433 + 0.900i)T^{2} \)
97 \( 1 + (0.781 - 0.623i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.680094753839869001170338855382, −8.701059012611473538275083391363, −7.37342964556104511925553611990, −6.57215665115214757650666216914, −5.54787844685853824725326265237, −4.79615253858860492259271741548, −4.24014837004915835384035470083, −3.41880989042496094579119782588, −2.53862250023522702512304287767, −1.79633969672301374502604461980, 2.31565470252632024713202170538, 2.81757457939640707320838845978, 3.92546505968028078171222820743, 4.58431077882121011618891905479, 5.81038789447162561491933846494, 6.10744160793023562204509443934, 7.15321135190117377133243291654, 7.82076184945436871951635080403, 8.022653821931747072885079396314, 9.155858153464336611362117221457

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