↑

Properties

Label 2-583-583.395-c0-0-0
Degree $2$
Conductor $583$
Sign $-0.990 - 0.140i$
Analytic cond. $0.290954$
Root an. cond. $0.539402$
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  + (−0.234 + 1.92i)3-s + (−0.748 + 0.663i)4-s + (−0.234 − 0.0576i)5-s + (−2.69 − 0.663i)9-s + (0.568 + 0.822i)11-s + (−1.10 − 1.59i)12-s + (0.166 − 0.437i)15-s + (0.120 − 0.992i)16-s + (0.213 − 0.112i)20-s + 1.13·23-s + (−0.833 − 0.437i)25-s + (1.21 − 3.21i)27-s + (−0.850 + 1.23i)31-s + (−1.71 + 0.902i)33-s + (2.45 − 1.28i)36-s + (−0.234 + 1.92i)37-s + ⋯
L(s)  = 1  + (−0.234 + 1.92i)3-s + (−0.748 + 0.663i)4-s + (−0.234 − 0.0576i)5-s + (−2.69 − 0.663i)9-s + (0.568 + 0.822i)11-s + (−1.10 − 1.59i)12-s + (0.166 − 0.437i)15-s + (0.120 − 0.992i)16-s + (0.213 − 0.112i)20-s + 1.13·23-s + (−0.833 − 0.437i)25-s + (1.21 − 3.21i)27-s + (−0.850 + 1.23i)31-s + (−1.71 + 0.902i)33-s + (2.45 − 1.28i)36-s + (−0.234 + 1.92i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(583\)    =    \(11 \cdot 53\)
Sign: $-0.990 - 0.140i$
Analytic conductor: \(0.290954\)
Root analytic conductor: \(0.539402\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{583} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 583,\ (\ :0),\ -0.990 - 0.140i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5914739964\)
\(L(\frac12)\) \(\approx\) \(0.5914739964\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.568 - 0.822i)T \)
53 \( 1 + (-0.885 + 0.464i)T \)
good2 \( 1 + (0.748 - 0.663i)T^{2} \)
3 \( 1 + (0.234 - 1.92i)T + (-0.970 - 0.239i)T^{2} \)
5 \( 1 + (0.234 + 0.0576i)T + (0.885 + 0.464i)T^{2} \)
7 \( 1 + (0.748 - 0.663i)T^{2} \)
13 \( 1 + (-0.120 - 0.992i)T^{2} \)
17 \( 1 + (-0.568 + 0.822i)T^{2} \)
19 \( 1 + (-0.120 - 0.992i)T^{2} \)
23 \( 1 - 1.13T + T^{2} \)
29 \( 1 + (0.354 + 0.935i)T^{2} \)
31 \( 1 + (0.850 - 1.23i)T + (-0.354 - 0.935i)T^{2} \)
37 \( 1 + (0.234 - 1.92i)T + (-0.970 - 0.239i)T^{2} \)
41 \( 1 + (0.354 - 0.935i)T^{2} \)
43 \( 1 + (0.970 - 0.239i)T^{2} \)
47 \( 1 + (0.234 - 0.0576i)T + (0.885 - 0.464i)T^{2} \)
59 \( 1 + (-1.45 + 0.358i)T + (0.885 - 0.464i)T^{2} \)
61 \( 1 + (-0.568 - 0.822i)T^{2} \)
67 \( 1 + (-0.530 + 0.470i)T + (0.120 - 0.992i)T^{2} \)
71 \( 1 + (-0.213 - 1.75i)T + (-0.970 + 0.239i)T^{2} \)
73 \( 1 + (-0.568 + 0.822i)T^{2} \)
79 \( 1 + (0.748 + 0.663i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-1.00 + 0.527i)T + (0.568 - 0.822i)T^{2} \)
97 \( 1 + (1.10 + 0.271i)T + (0.885 + 0.464i)T^{2} \)
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

−11.30975233254825421742923057609, −10.20188450341541867219825875570, −9.655583411909095489092571976742, −8.895883557705346433519521960404, −8.230171604443616571681025609898, −6.82117774005337625022008979546, −5.35108251501244647746693734985, −4.67774983876924357700889168973, −3.92195907132209964594225551508, −3.08297576565633051422130339279, 0.75536240035333267635461654937, 2.05394836432634076337249091427, 3.66344719927573111035646925335, 5.36404876749764760871217229360, 5.96215841754000630818014536492, 6.89823140917987804637869137957, 7.75170127523825343206870068966, 8.649484085191389404703923106590, 9.334491570525760047753591076806, 10.86857381141508049374142573928

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