Properties

Label 2-2001-2001.965-c0-0-4
Degree $2$
Conductor $2001$
Sign $-0.117 + 0.993i$
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  + (0.500 − 1.43i)2-s + (0.955 + 0.294i)3-s + (−1.01 − 0.809i)4-s + (0.900 − 1.21i)6-s + (−0.382 + 0.240i)8-s + (0.826 + 0.563i)9-s + (−0.731 − 1.07i)12-s + (0.145 − 0.0332i)13-s + (−0.136 − 0.597i)16-s + (1.21 − 0.900i)18-s + (−0.433 + 0.900i)23-s + (−0.436 + 0.116i)24-s + (0.623 − 0.781i)25-s + (0.0253 − 0.225i)26-s + (0.623 + 0.781i)27-s + ⋯
L(s)  = 1  + (0.500 − 1.43i)2-s + (0.955 + 0.294i)3-s + (−1.01 − 0.809i)4-s + (0.900 − 1.21i)6-s + (−0.382 + 0.240i)8-s + (0.826 + 0.563i)9-s + (−0.731 − 1.07i)12-s + (0.145 − 0.0332i)13-s + (−0.136 − 0.597i)16-s + (1.21 − 0.900i)18-s + (−0.433 + 0.900i)23-s + (−0.436 + 0.116i)24-s + (0.623 − 0.781i)25-s + (0.0253 − 0.225i)26-s + (0.623 + 0.781i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.117 + 0.993i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (965, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ -0.117 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.085627347\)
\(L(\frac12)\) \(\approx\) \(2.085627347\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.955 - 0.294i)T \)
23 \( 1 + (0.433 - 0.900i)T \)
29 \( 1 + (0.930 - 0.365i)T \)
good2 \( 1 + (-0.500 + 1.43i)T + (-0.781 - 0.623i)T^{2} \)
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 \( 1 + (-0.145 + 0.0332i)T + (0.900 - 0.433i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (-0.974 + 0.222i)T^{2} \)
31 \( 1 + (-0.488 - 0.170i)T + (0.781 + 0.623i)T^{2} \)
37 \( 1 + (0.433 - 0.900i)T^{2} \)
41 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
43 \( 1 + (0.781 - 0.623i)T^{2} \)
47 \( 1 + (-0.425 + 0.677i)T + (-0.433 - 0.900i)T^{2} \)
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 \( 1 + (0.250 + 1.09i)T + (-0.900 + 0.433i)T^{2} \)
73 \( 1 + (1.12 - 0.392i)T + (0.781 - 0.623i)T^{2} \)
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

−9.270891230145568319129169882278, −8.695946928578710601530396189346, −7.70573795810221583867576403822, −6.93922988761389969672249006831, −5.57309033284444666799418765423, −4.67722719510831442053343709612, −3.89187480691116393684264467318, −3.23094730540402999340392806298, −2.35320469836463540503668349565, −1.45365314213183018153841147944, 1.72143489761037131323511587508, 3.00968072097131036704871116275, 4.00658326497134886431990884450, 4.76254252664900855908989605708, 5.74374701516961313995774824322, 6.62087860552018012339193420297, 7.08212643135279605374218308506, 8.018535315543953503518885521982, 8.363587328674148115182109340206, 9.219504305068013795673770723728

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