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Properties

Label 2-2001-2001.68-c0-0-5
Degree $2$
Conductor $2001$
Sign $0.583 - 0.811i$
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  + (−0.940 − 1.49i)2-s + (0.365 − 0.930i)3-s + (−0.922 + 1.91i)4-s + (−1.73 + 0.328i)6-s + (1.97 − 0.223i)8-s + (−0.733 − 0.680i)9-s + (1.44 + 1.55i)12-s + (−1.49 + 1.19i)13-s + (−0.870 − 1.09i)16-s + (−0.328 + 1.73i)18-s + (−0.974 + 0.222i)23-s + (0.515 − 1.92i)24-s + (−0.900 − 0.433i)25-s + (3.18 + 1.11i)26-s + (−0.900 + 0.433i)27-s + ⋯
L(s)  = 1  + (−0.940 − 1.49i)2-s + (0.365 − 0.930i)3-s + (−0.922 + 1.91i)4-s + (−1.73 + 0.328i)6-s + (1.97 − 0.223i)8-s + (−0.733 − 0.680i)9-s + (1.44 + 1.55i)12-s + (−1.49 + 1.19i)13-s + (−0.870 − 1.09i)16-s + (−0.328 + 1.73i)18-s + (−0.974 + 0.222i)23-s + (0.515 − 1.92i)24-s + (−0.900 − 0.433i)25-s + (3.18 + 1.11i)26-s + (−0.900 + 0.433i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.583 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.583 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.583 - 0.811i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ 0.583 - 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04186640562\)
\(L(\frac12)\) \(\approx\) \(0.04186640562\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.365 + 0.930i)T \)
23 \( 1 + (0.974 - 0.222i)T \)
29 \( 1 + (0.997 - 0.0747i)T \)
good2 \( 1 + (0.940 + 1.49i)T + (-0.433 + 0.900i)T^{2} \)
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 \( 1 + (1.49 - 1.19i)T + (0.222 - 0.974i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (0.781 - 0.623i)T^{2} \)
31 \( 1 + (1.63 - 1.02i)T + (0.433 - 0.900i)T^{2} \)
37 \( 1 + (0.974 - 0.222i)T^{2} \)
41 \( 1 + (1.13 + 1.13i)T + iT^{2} \)
43 \( 1 + (0.433 + 0.900i)T^{2} \)
47 \( 1 + (0.146 - 1.29i)T + (-0.974 - 0.222i)T^{2} \)
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 \( 1 + (0.848 + 1.06i)T + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (-1.66 - 1.04i)T + (0.433 + 0.900i)T^{2} \)
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

−8.966773774943360938417971281387, −8.050689177634808124458747422189, −7.44715919342808606952965699640, −6.69799180628157628522521319953, −5.39744855275496769926382750127, −4.08328917755737076561215057268, −3.29796747174548773640530278110, −2.04997489082243683645897606359, −1.88102875090444196826947201403, −0.03593774278688144680402585589, 2.20520346768547421036099751334, 3.57215515097785888242396645089, 4.68479734254333650802491527454, 5.50371664932196288400998112557, 5.91862702759726904536501634057, 7.29990338831577914067343363442, 7.65633606206493073783767688409, 8.373129609594222484459690519565, 9.166460863249492302300401563949, 9.856662107611102886443681625485

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