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Properties

Label 2-2001-2001.482-c0-0-3
Degree $2$
Conductor $2001$
Sign $0.995 - 0.0975i$
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  + (−1.23 + 0.430i)2-s + (0.955 − 0.294i)3-s + (0.548 − 0.437i)4-s + (−1.04 + 0.774i)6-s + (0.206 − 0.329i)8-s + (0.826 − 0.563i)9-s + (0.395 − 0.579i)12-s + (−0.145 − 0.0332i)13-s + (−0.269 + 1.17i)16-s + (−0.774 + 1.04i)18-s + (0.433 + 0.900i)23-s + (0.100 − 0.375i)24-s + (0.623 + 0.781i)25-s + (0.193 − 0.0218i)26-s + (0.623 − 0.781i)27-s + ⋯
L(s)  = 1  + (−1.23 + 0.430i)2-s + (0.955 − 0.294i)3-s + (0.548 − 0.437i)4-s + (−1.04 + 0.774i)6-s + (0.206 − 0.329i)8-s + (0.826 − 0.563i)9-s + (0.395 − 0.579i)12-s + (−0.145 − 0.0332i)13-s + (−0.269 + 1.17i)16-s + (−0.774 + 1.04i)18-s + (0.433 + 0.900i)23-s + (0.100 − 0.375i)24-s + (0.623 + 0.781i)25-s + (0.193 − 0.0218i)26-s + (0.623 − 0.781i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.995 - 0.0975i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (482, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ 0.995 - 0.0975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8787684253\)
\(L(\frac12)\) \(\approx\) \(0.8787684253\)
\(L(1)\) 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 \( 1 + (1.23 - 0.430i)T + (0.781 - 0.623i)T^{2} \)
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 \( 1 + (0.638 + 1.82i)T + (-0.781 + 0.623i)T^{2} \)
37 \( 1 + (-0.433 - 0.900i)T^{2} \)
41 \( 1 + (0.0528 + 0.0528i)T + iT^{2} \)
43 \( 1 + (-0.781 - 0.623i)T^{2} \)
47 \( 1 + (-1.55 + 0.975i)T + (0.433 - 0.900i)T^{2} \)
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 \( 1 + (0.531 - 1.51i)T + (-0.781 - 0.623i)T^{2} \)
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

−9.117475024882819152304175256563, −8.691132653883559764422194016419, −7.83946828211896553162314607777, −7.30216500942508417254717997818, −6.71371352675477044040103535630, −5.58114430448665912746829954117, −4.28961647964216462121322592481, −3.41697622068782515231729780335, −2.22845905943360691840068214745, −1.08856002052976390898478681196, 1.21234371746410461182917854769, 2.37626303505615750128156043625, 3.08146783089238185738369067901, 4.38890751108161602491544703441, 5.07512394978233146226190278634, 6.52584263338550186213811277110, 7.36469976621666421647836970518, 8.102242598233119517009002830856, 8.769932066155268086334487448446, 9.150262374089477757323648030246

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