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Properties

Label 2-2001-2001.620-c0-0-2
Degree $2$
Conductor $2001$
Sign $-0.801 + 0.598i$
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  + (−0.623 − 1.78i)2-s + (−0.222 + 0.974i)3-s + (−2.00 + 1.59i)4-s + (1.87 − 0.211i)6-s + (2.49 + 1.57i)8-s + (−0.900 − 0.433i)9-s + (−1.11 − 2.30i)12-s + (−1.75 − 0.400i)13-s + (0.669 − 2.93i)16-s + (−0.211 + 1.87i)18-s + (−0.433 − 0.900i)23-s + (−2.08 + 2.08i)24-s + (0.623 + 0.781i)25-s + (0.380 + 3.38i)26-s + (0.623 − 0.781i)27-s + ⋯
L(s)  = 1  + (−0.623 − 1.78i)2-s + (−0.222 + 0.974i)3-s + (−2.00 + 1.59i)4-s + (1.87 − 0.211i)6-s + (2.49 + 1.57i)8-s + (−0.900 − 0.433i)9-s + (−1.11 − 2.30i)12-s + (−1.75 − 0.400i)13-s + (0.669 − 2.93i)16-s + (−0.211 + 1.87i)18-s + (−0.433 − 0.900i)23-s + (−2.08 + 2.08i)24-s + (0.623 + 0.781i)25-s + (0.380 + 3.38i)26-s + (0.623 − 0.781i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.801 + 0.598i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (620, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ -0.801 + 0.598i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4522588296\)
\(L(\frac12)\) \(\approx\) \(0.4522588296\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.222 - 0.974i)T \)
23 \( 1 + (0.433 + 0.900i)T \)
29 \( 1 + (-0.781 + 0.623i)T \)
good2 \( 1 + (0.623 + 1.78i)T + (-0.781 + 0.623i)T^{2} \)
5 \( 1 + (-0.623 - 0.781i)T^{2} \)
7 \( 1 + (0.222 + 0.974i)T^{2} \)
11 \( 1 + (-0.433 + 0.900i)T^{2} \)
13 \( 1 + (1.75 + 0.400i)T + (0.900 + 0.433i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (-0.974 - 0.222i)T^{2} \)
31 \( 1 + (-1.33 + 0.467i)T + (0.781 - 0.623i)T^{2} \)
37 \( 1 + (0.433 + 0.900i)T^{2} \)
41 \( 1 + (-0.752 + 0.752i)T - iT^{2} \)
43 \( 1 + (0.781 + 0.623i)T^{2} \)
47 \( 1 + (1.05 + 1.68i)T + (-0.433 + 0.900i)T^{2} \)
53 \( 1 + (0.623 + 0.781i)T^{2} \)
59 \( 1 + 1.24iT - T^{2} \)
61 \( 1 + (0.974 - 0.222i)T^{2} \)
67 \( 1 + (-0.900 + 0.433i)T^{2} \)
71 \( 1 + (0.193 - 0.846i)T + (-0.900 - 0.433i)T^{2} \)
73 \( 1 + (-1.87 - 0.656i)T + (0.781 + 0.623i)T^{2} \)
79 \( 1 + (-0.433 - 0.900i)T^{2} \)
83 \( 1 + (-0.222 + 0.974i)T^{2} \)
89 \( 1 + (0.781 - 0.623i)T^{2} \)
97 \( 1 + (0.974 + 0.222i)T^{2} \)
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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

−9.579871726088194199343541070784, −8.516529720050788545303150837436, −8.057157376613406506727143770895, −6.79322644173615937281670663981, −5.29453146117627871355035786781, −4.68798217795589366467592404820, −3.89505425516310588510056790561, −2.90724172506648742573410154711, −2.27338083380008952489515309338, −0.47403144761599708117908831176, 1.17947667341555740482404024951, 2.69325925309115995567696262578, 4.61893769783894818744984020336, 5.04530858755335073485963135015, 6.15361131031997115697961724863, 6.55792127718321128352289933756, 7.40040542694339893679835513968, 7.81596627801746528099906010979, 8.543146676286509077783810512755, 9.404769899563263248644833179807

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