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Properties

Label 2-2883-93.8-c0-0-6
Degree $2$
Conductor $2883$
Sign $0.0752 + 0.997i$
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  + (0.951 + 0.309i)2-s + (0.453 − 0.891i)3-s − i·5-s + (0.707 − 0.707i)6-s + (0.809 + 0.587i)7-s + (−0.587 − 0.809i)8-s + (−0.587 − 0.809i)9-s + (0.309 − 0.951i)10-s + (0.587 + 0.809i)14-s + (−0.891 − 0.453i)15-s + (−0.309 − 0.951i)16-s + (0.831 + 1.14i)17-s + (−0.309 − 0.951i)18-s + (0.309 − 0.951i)19-s + (0.891 − 0.453i)21-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)2-s + (0.453 − 0.891i)3-s − i·5-s + (0.707 − 0.707i)6-s + (0.809 + 0.587i)7-s + (−0.587 − 0.809i)8-s + (−0.587 − 0.809i)9-s + (0.309 − 0.951i)10-s + (0.587 + 0.809i)14-s + (−0.891 − 0.453i)15-s + (−0.309 − 0.951i)16-s + (0.831 + 1.14i)17-s + (−0.309 − 0.951i)18-s + (0.309 − 0.951i)19-s + (0.891 − 0.453i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0752 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0752 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2883\)    =    \(3 \cdot 31^{2}\)
Sign: $0.0752 + 0.997i$
Analytic conductor: \(1.43880\)
Root analytic conductor: \(1.19950\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2883} (1589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2883,\ (\ :0),\ 0.0752 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.198196972\)
\(L(\frac12)\) \(\approx\) \(2.198196972\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.453 + 0.891i)T \)
31 \( 1 \)
good2 \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \)
5 \( 1 + iT - T^{2} \)
7 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
11 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.831 - 1.14i)T + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (1.34 + 0.437i)T + (0.809 + 0.587i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \)
43 \( 1 + (0.437 - 1.34i)T + (-0.809 - 0.587i)T^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
61 \( 1 - 1.41T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (1.34 + 0.437i)T + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (-0.309 - 0.951i)T^{2} \)
97 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
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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

−8.595108785079460221605944326031, −8.112629270844891560316066526071, −7.19572798248096167243057571377, −6.33819194192278041554242564735, −5.51013577304456418238162499889, −5.12267812142190632907088364196, −4.12710642793995584595047045811, −3.26812180278011093091308842808, −2.04177907635318126066102834741, −1.03500000886511421970374054109, 1.98250557323724143719386622107, 3.03625784569626993908862257455, 3.54584083045496564730379019852, 4.26040985071515780223231614058, 5.20272005901193215366972590022, 5.57157742432291243374935901767, 6.92437892693810631146637404501, 7.67866285491807076170336612257, 8.333772572032062358581153141962, 9.205182876801631863322374053779

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