↑

Properties

Label 2-2008-2008.155-c0-0-0
Degree $2$
Conductor $2008$
Sign $-0.494 + 0.869i$
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  + (−0.992 − 0.125i)2-s + (−0.507 − 0.430i)3-s + (0.968 + 0.248i)4-s + (0.449 + 0.490i)6-s + (−0.929 − 0.368i)8-s + (−0.0905 − 0.549i)9-s + (1.17 − 1.57i)11-s + (−0.384 − 0.543i)12-s + (0.876 + 0.481i)16-s + (0.0697 − 0.614i)17-s + (0.0209 + 0.556i)18-s + (−0.847 − 0.0854i)19-s + (−1.36 + 1.41i)22-s + (0.313 + 0.587i)24-s + (−0.425 + 0.904i)25-s + ⋯
L(s)  = 1  + (−0.992 − 0.125i)2-s + (−0.507 − 0.430i)3-s + (0.968 + 0.248i)4-s + (0.449 + 0.490i)6-s + (−0.929 − 0.368i)8-s + (−0.0905 − 0.549i)9-s + (1.17 − 1.57i)11-s + (−0.384 − 0.543i)12-s + (0.876 + 0.481i)16-s + (0.0697 − 0.614i)17-s + (0.0209 + 0.556i)18-s + (−0.847 − 0.0854i)19-s + (−1.36 + 1.41i)22-s + (0.313 + 0.587i)24-s + (−0.425 + 0.904i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.494 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.494 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2008\)    =    \(2^{3} \cdot 251\)
Sign: $-0.494 + 0.869i$
Analytic conductor: \(1.00212\)
Root analytic conductor: \(1.00106\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2008} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2008,\ (\ :0),\ -0.494 + 0.869i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5485456888\)
\(L(\frac12)\) \(\approx\) \(0.5485456888\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.992 + 0.125i)T \)
251 \( 1 + (0.0878 - 0.996i)T \)
good3 \( 1 + (0.507 + 0.430i)T + (0.162 + 0.986i)T^{2} \)
5 \( 1 + (0.425 - 0.904i)T^{2} \)
7 \( 1 + (0.984 - 0.175i)T^{2} \)
11 \( 1 + (-1.17 + 1.57i)T + (-0.285 - 0.958i)T^{2} \)
13 \( 1 + (-0.954 - 0.297i)T^{2} \)
17 \( 1 + (-0.0697 + 0.614i)T + (-0.974 - 0.224i)T^{2} \)
19 \( 1 + (0.847 + 0.0854i)T + (0.979 + 0.199i)T^{2} \)
23 \( 1 + (0.514 - 0.857i)T^{2} \)
29 \( 1 + (-0.850 + 0.525i)T^{2} \)
31 \( 1 + (-0.793 - 0.607i)T^{2} \)
37 \( 1 + (0.597 + 0.801i)T^{2} \)
41 \( 1 + (-1.91 - 0.341i)T + (0.938 + 0.344i)T^{2} \)
43 \( 1 + (1.44 + 1.10i)T + (0.260 + 0.965i)T^{2} \)
47 \( 1 + (-0.535 + 0.844i)T^{2} \)
53 \( 1 + (-0.356 - 0.934i)T^{2} \)
59 \( 1 + (0.213 + 1.88i)T + (-0.974 + 0.224i)T^{2} \)
61 \( 1 + (0.778 - 0.627i)T^{2} \)
67 \( 1 + (0.111 + 0.798i)T + (-0.962 + 0.272i)T^{2} \)
71 \( 1 + (-0.577 + 0.816i)T^{2} \)
73 \( 1 + (-0.373 + 0.0282i)T + (0.988 - 0.150i)T^{2} \)
79 \( 1 + (0.236 - 0.971i)T^{2} \)
83 \( 1 + (-0.636 + 0.415i)T + (0.402 - 0.915i)T^{2} \)
89 \( 1 + (1.09 - 0.222i)T + (0.920 - 0.391i)T^{2} \)
97 \( 1 + (-0.800 - 0.645i)T + (0.212 + 0.977i)T^{2} \)
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

−9.142223740897448452734148657042, −8.427142433548474604168750427431, −7.59435660766409816693802077293, −6.57302047820772995711527806008, −6.33059259347894779487977570882, −5.41654925710082574292477995035, −3.82263395263583198791704724009, −3.11443913285434972131442309363, −1.68100763865612885233285622558, −0.59922179356326253630681100114, 1.57153604037388013916186382318, 2.44432759038562712151708349303, 4.02925924404684247521160760532, 4.72270640471719863889776517488, 5.90494397063082467785545901936, 6.49343029879953099815132528595, 7.34511759149933287975177304726, 8.096944931406596586665876139323, 8.888216387597205754507016016495, 9.769330150925502020751141923064

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