↑

Properties

Label 2-2883-93.2-c0-0-6
Degree $2$
Conductor $2883$
Sign $0.0679 + 0.997i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.587 − 0.809i)2-s + (0.987 + 0.156i)3-s − i·5-s + (0.707 − 0.707i)6-s + (−0.309 − 0.951i)7-s + (0.951 + 0.309i)8-s + (0.951 + 0.309i)9-s + (−0.809 − 0.587i)10-s + (−0.951 − 0.309i)14-s + (0.156 − 0.987i)15-s + (0.809 − 0.587i)16-s + (−1.34 − 0.437i)17-s + (0.809 − 0.587i)18-s + (−0.809 − 0.587i)19-s + (−0.156 − 0.987i)21-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)2-s + (0.987 + 0.156i)3-s − i·5-s + (0.707 − 0.707i)6-s + (−0.309 − 0.951i)7-s + (0.951 + 0.309i)8-s + (0.951 + 0.309i)9-s + (−0.809 − 0.587i)10-s + (−0.951 − 0.309i)14-s + (0.156 − 0.987i)15-s + (0.809 − 0.587i)16-s + (−1.34 − 0.437i)17-s + (0.809 − 0.587i)18-s + (−0.809 − 0.587i)19-s + (−0.156 − 0.987i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0679 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0679 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2883\)    =    \(3 \cdot 31^{2}\)
Sign: $0.0679 + 0.997i$
Analytic conductor: \(1.43880\)
Root analytic conductor: \(1.19950\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2883} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2883,\ (\ :0),\ 0.0679 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.443218074\)
\(L(\frac12)\) \(\approx\) \(2.443218074\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.987 - 0.156i)T \)
31 \( 1 \)
good2 \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \)
5 \( 1 + iT - T^{2} \)
7 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
11 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (1.34 + 0.437i)T + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.809 + 0.587i)T^{2} \)
29 \( 1 + (0.831 - 1.14i)T + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \)
43 \( 1 + (-1.14 - 0.831i)T + (0.309 + 0.951i)T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
61 \( 1 - 1.41T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.831 - 1.14i)T + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.802345305705487215678766709457, −8.211772785494857549584428482871, −7.26520080998335471852560156585, −6.78858400562009922710639516819, −5.22506965389907053494097040850, −4.39413862136068273368128873009, −4.13505014783621425985169541894, −3.11690995770040174868700796082, −2.28573136964910840486222237685, −1.25011863822392860215988094231, 1.97674474837577383446159896838, 2.50997092027536055852758014939, 3.71562089069994889901614722284, 4.31284700711668678194780419240, 5.52554729101555492558961886711, 6.24423942267773880148309145503, 6.79524521181609112729052455540, 7.41478590684818617199936918419, 8.309242272830925195579295687153, 8.923238288203348473499469675123

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