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Properties

Label 2-2883-93.5-c0-0-3
Degree $2$
Conductor $2883$
Sign $0.992 + 0.122i$
Analytic cond. $1.43880$
Root an. cond. $1.19950$
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  + (−0.5 + 0.866i)3-s + 4-s + (0.809 − 1.40i)7-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)12-s + (−0.309 − 0.535i)13-s + 16-s + (−0.309 + 0.535i)19-s + (0.809 + 1.40i)21-s + (−0.5 + 0.866i)25-s + 0.999·27-s + (0.809 − 1.40i)28-s + (−0.499 − 0.866i)36-s + (0.809 − 1.40i)37-s + 0.618·39-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + 4-s + (0.809 − 1.40i)7-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)12-s + (−0.309 − 0.535i)13-s + 16-s + (−0.309 + 0.535i)19-s + (0.809 + 1.40i)21-s + (−0.5 + 0.866i)25-s + 0.999·27-s + (0.809 − 1.40i)28-s + (−0.499 − 0.866i)36-s + (0.809 − 1.40i)37-s + 0.618·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2883\)    =    \(3 \cdot 31^{2}\)
Sign: $0.992 + 0.122i$
Analytic conductor: \(1.43880\)
Root analytic conductor: \(1.19950\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2883} (1400, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2883,\ (\ :0),\ 0.992 + 0.122i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.415519460\)
\(L(\frac12)\) \(\approx\) \(1.415519460\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 \)
good2 \( 1 - T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
37 \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 - 0.618T + T^{2} \)
67 \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - 0.618T + T^{2} \)
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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

−9.031902985223425201659401510145, −7.945131601134113771814084126758, −7.46792800319782722097140503840, −6.69931037229540219207404308177, −5.76694055033963290845533968000, −5.15309575916397084711536952787, −4.07149374860187121862846995188, −3.60694314831453334431396532248, −2.31225416487307896512662421530, −1.02283247813362005886739235994, 1.44535510677861701219142717230, 2.28787292986118062031402043478, 2.81352606549824779339387564220, 4.52420817851299238975475776843, 5.30306826729075552342853556042, 6.14663881895642028648574357975, 6.49607145111873435323561229592, 7.49932502682565592683292746948, 8.064865600397923895367360207027, 8.724546433274864607327842958804

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