↑

Properties

Label 2-2883-93.47-c0-0-2
Degree $2$
Conductor $2883$
Sign $-0.600 - 0.799i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2883\)    =    \(3 \cdot 31^{2}\)
Sign: $-0.600 - 0.799i$
Analytic conductor: \(1.43880\)
Root analytic conductor: \(1.19950\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2883} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2883,\ (\ :0),\ -0.600 - 0.799i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.312559733\)
\(L(\frac12)\) \(\approx\) \(1.312559733\)
\(L(1)\) 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

−9.348705886944444119207035911289, −8.208442035977427464649513304759, −7.32913875162808028426657647708, −6.63708500812796932603645630728, −6.23838183110058411753216656737, −5.44886699275347281703958174908, −4.97478915155645885752054690145, −3.85178027290348078846983815803, −2.91140020813254785966082064812, −1.51637632850506793098263349133, 0.852123042272299024696663669984, 1.76697255552213091308616824345, 3.14722759232583562246947820594, 4.13753725247929708592135180625, 4.59729844122965067519440141612, 5.35386384059655142722432283777, 6.29073376396062809611305759809, 7.14521291762246153997093021732, 7.86586889695030691897699108330, 8.610327449407491786265739060345

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