Properties

Label 2-2008-2008.1459-c0-0-0
Degree $2$
Conductor $2008$
Sign $0.709 - 0.704i$
Analytic cond. $1.00212$
Root an. cond. $1.00106$
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  + (0.309 + 0.951i)2-s + (−1.11 + 0.614i)3-s + (−0.809 + 0.587i)4-s + (−0.929 − 0.872i)6-s + (−0.809 − 0.587i)8-s + (0.335 − 0.527i)9-s + (0.844 − 1.79i)11-s + (0.542 − 1.15i)12-s + (0.309 − 0.951i)16-s + (1.45 − 1.36i)17-s + (0.605 + 0.155i)18-s + (−0.574 − 0.227i)19-s + (1.96 + 0.248i)22-s + (1.26 + 0.159i)24-s + (0.309 + 0.951i)25-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (−1.11 + 0.614i)3-s + (−0.809 + 0.587i)4-s + (−0.929 − 0.872i)6-s + (−0.809 − 0.587i)8-s + (0.335 − 0.527i)9-s + (0.844 − 1.79i)11-s + (0.542 − 1.15i)12-s + (0.309 − 0.951i)16-s + (1.45 − 1.36i)17-s + (0.605 + 0.155i)18-s + (−0.574 − 0.227i)19-s + (1.96 + 0.248i)22-s + (1.26 + 0.159i)24-s + (0.309 + 0.951i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2008\)    =    \(2^{3} \cdot 251\)
Sign: $0.709 - 0.704i$
Analytic conductor: \(1.00212\)
Root analytic conductor: \(1.00106\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2008} (1459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2008,\ (\ :0),\ 0.709 - 0.704i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8012952184\)
\(L(\frac12)\) \(\approx\) \(0.8012952184\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
251 \( 1 + (-0.0627 - 0.998i)T \)
good3 \( 1 + (1.11 - 0.614i)T + (0.535 - 0.844i)T^{2} \)
5 \( 1 + (-0.309 - 0.951i)T^{2} \)
7 \( 1 + (0.992 + 0.125i)T^{2} \)
11 \( 1 + (-0.844 + 1.79i)T + (-0.637 - 0.770i)T^{2} \)
13 \( 1 + (0.425 + 0.904i)T^{2} \)
17 \( 1 + (-1.45 + 1.36i)T + (0.0627 - 0.998i)T^{2} \)
19 \( 1 + (0.574 + 0.227i)T + (0.728 + 0.684i)T^{2} \)
23 \( 1 + (-0.0627 + 0.998i)T^{2} \)
29 \( 1 + (-0.876 + 0.481i)T^{2} \)
31 \( 1 + (0.637 + 0.770i)T^{2} \)
37 \( 1 + (0.425 + 0.904i)T^{2} \)
41 \( 1 + (0.124 - 0.0157i)T + (0.968 - 0.248i)T^{2} \)
43 \( 1 + (0.929 + 1.12i)T + (-0.187 + 0.982i)T^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.187 + 0.982i)T^{2} \)
59 \( 1 + (0.620 + 0.582i)T + (0.0627 + 0.998i)T^{2} \)
61 \( 1 + (0.187 + 0.982i)T^{2} \)
67 \( 1 + (-1.84 + 0.233i)T + (0.968 - 0.248i)T^{2} \)
71 \( 1 + (0.425 + 0.904i)T^{2} \)
73 \( 1 + (-0.541 - 0.297i)T + (0.535 + 0.844i)T^{2} \)
79 \( 1 + (-0.876 - 0.481i)T^{2} \)
83 \( 1 + (0.328 - 1.72i)T + (-0.929 - 0.368i)T^{2} \)
89 \( 1 + (-1.41 + 1.32i)T + (0.0627 - 0.998i)T^{2} \)
97 \( 1 + (0.0235 - 0.123i)T + (-0.929 - 0.368i)T^{2} \)
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

−9.360754808520031756573780138087, −8.615175979079371895043589388858, −7.81750671167175299791173222956, −6.78730055171444746318124966873, −6.18960289759680990998809706124, −5.36031250895516631435375596157, −5.04038624538058854533161908462, −3.81328905618493481575623810250, −3.17186764926558004538949443982, −0.71484707269729334135984375663, 1.25954878001978811915229432244, 1.98569930727970895796731720164, 3.46510803943351517333332388891, 4.39877521942929973163124090671, 5.10837487279901156511619513572, 6.13256383748206983812574484326, 6.53475858438595011968736086031, 7.63943526032982630940466411942, 8.542856740174160272366758279506, 9.636781980409802850790558051753

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