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Properties

Label 2-2001-2001.827-c0-0-3
Degree $2$
Conductor $2001$
Sign $0.129 - 0.991i$
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  + (0.0895 + 0.794i)2-s + (0.930 + 0.365i)3-s + (0.351 − 0.0801i)4-s + (−0.206 + 0.772i)6-s + (0.359 + 1.02i)8-s + (0.733 + 0.680i)9-s + (0.356 + 0.0537i)12-s + (−0.317 − 0.658i)13-s + (−0.459 + 0.221i)16-s + (−0.474 + 0.643i)18-s + (−0.781 + 0.623i)23-s + (−0.0406 + 1.08i)24-s + (−0.222 − 0.974i)25-s + (0.494 − 0.310i)26-s + (0.433 + 0.900i)27-s + ⋯
L(s)  = 1  + (0.0895 + 0.794i)2-s + (0.930 + 0.365i)3-s + (0.351 − 0.0801i)4-s + (−0.206 + 0.772i)6-s + (0.359 + 1.02i)8-s + (0.733 + 0.680i)9-s + (0.356 + 0.0537i)12-s + (−0.317 − 0.658i)13-s + (−0.459 + 0.221i)16-s + (−0.474 + 0.643i)18-s + (−0.781 + 0.623i)23-s + (−0.0406 + 1.08i)24-s + (−0.222 − 0.974i)25-s + (0.494 − 0.310i)26-s + (0.433 + 0.900i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.129 - 0.991i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ 0.129 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.935464810\)
\(L(\frac12)\) \(\approx\) \(1.935464810\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.930 - 0.365i)T \)
23 \( 1 + (0.781 - 0.623i)T \)
29 \( 1 + (-0.294 + 0.955i)T \)
good2 \( 1 + (-0.0895 - 0.794i)T + (-0.974 + 0.222i)T^{2} \)
5 \( 1 + (0.222 + 0.974i)T^{2} \)
7 \( 1 + (0.900 + 0.433i)T^{2} \)
11 \( 1 + (0.781 + 0.623i)T^{2} \)
13 \( 1 + (0.317 + 0.658i)T + (-0.623 + 0.781i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (0.433 + 0.900i)T^{2} \)
31 \( 1 + (0.514 - 0.0579i)T + (0.974 - 0.222i)T^{2} \)
37 \( 1 + (-0.781 + 0.623i)T^{2} \)
41 \( 1 + (0.262 + 0.262i)T + iT^{2} \)
43 \( 1 + (0.974 + 0.222i)T^{2} \)
47 \( 1 + (0.882 + 0.308i)T + (0.781 + 0.623i)T^{2} \)
53 \( 1 + (-0.222 - 0.974i)T^{2} \)
59 \( 1 - 0.445iT - T^{2} \)
61 \( 1 + (-0.433 + 0.900i)T^{2} \)
67 \( 1 + (0.623 + 0.781i)T^{2} \)
71 \( 1 + (0.268 - 0.129i)T + (0.623 - 0.781i)T^{2} \)
73 \( 1 + (-1.98 - 0.223i)T + (0.974 + 0.222i)T^{2} \)
79 \( 1 + (0.781 - 0.623i)T^{2} \)
83 \( 1 + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (0.974 - 0.222i)T^{2} \)
97 \( 1 + (-0.433 - 0.900i)T^{2} \)
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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

−9.518476913582888274909925279139, −8.354248543761749242228301683577, −8.056621392625088583249269912003, −7.29697774616394310745614137264, −6.47171437572684496900239196000, −5.57490471289498819598013749560, −4.77716688329707768002747145938, −3.77297846455488949182911135650, −2.71553124748597102027591853365, −1.86228680574697451805070332852, 1.45449903409184651685153752049, 2.21434834782035697371502439460, 3.16816740357971050834743426257, 3.85874147232107952170180121178, 4.84115753000580399426579446011, 6.27757938779676813422700507286, 6.91408508874069617950414188442, 7.63034915885225664360731363799, 8.404748929617574863596412777863, 9.368103241126428650324055712618

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