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Properties

Label 2-2008-2008.1355-c0-0-0
Degree $2$
Conductor $2008$
Sign $0.735 - 0.677i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 − 0.587i)2-s + (−0.929 + 1.12i)3-s + (0.309 + 0.951i)4-s + (1.41 − 0.362i)6-s + (0.309 − 0.951i)8-s + (−0.210 − 1.10i)9-s + (−0.996 − 0.394i)11-s + (−1.35 − 0.536i)12-s + (−0.809 + 0.587i)16-s + (1.93 + 0.497i)17-s + (−0.479 + 1.01i)18-s + (1.60 − 0.202i)19-s + (0.574 + 0.904i)22-s + (0.781 + 1.23i)24-s + (−0.809 − 0.587i)25-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (−0.929 + 1.12i)3-s + (0.309 + 0.951i)4-s + (1.41 − 0.362i)6-s + (0.309 − 0.951i)8-s + (−0.210 − 1.10i)9-s + (−0.996 − 0.394i)11-s + (−1.35 − 0.536i)12-s + (−0.809 + 0.587i)16-s + (1.93 + 0.497i)17-s + (−0.479 + 1.01i)18-s + (1.60 − 0.202i)19-s + (0.574 + 0.904i)22-s + (0.781 + 1.23i)24-s + (−0.809 − 0.587i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2008\)    =    \(2^{3} \cdot 251\)
Sign: $0.735 - 0.677i$
Analytic conductor: \(1.00212\)
Root analytic conductor: \(1.00106\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2008} (1355, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2008,\ (\ :0),\ 0.735 - 0.677i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5535180674\)
\(L(\frac12)\) \(\approx\) \(0.5535180674\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
251 \( 1 + (-0.876 + 0.481i)T \)
good3 \( 1 + (0.929 - 1.12i)T + (-0.187 - 0.982i)T^{2} \)
5 \( 1 + (0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.535 - 0.844i)T^{2} \)
11 \( 1 + (0.996 + 0.394i)T + (0.728 + 0.684i)T^{2} \)
13 \( 1 + (0.929 - 0.368i)T^{2} \)
17 \( 1 + (-1.93 - 0.497i)T + (0.876 + 0.481i)T^{2} \)
19 \( 1 + (-1.60 + 0.202i)T + (0.968 - 0.248i)T^{2} \)
23 \( 1 + (-0.876 - 0.481i)T^{2} \)
29 \( 1 + (0.637 - 0.770i)T^{2} \)
31 \( 1 + (-0.728 - 0.684i)T^{2} \)
37 \( 1 + (0.929 - 0.368i)T^{2} \)
41 \( 1 + (-0.939 + 1.47i)T + (-0.425 - 0.904i)T^{2} \)
43 \( 1 + (-1.41 - 1.32i)T + (0.0627 + 0.998i)T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.0627 + 0.998i)T^{2} \)
59 \( 1 + (1.80 - 0.462i)T + (0.876 - 0.481i)T^{2} \)
61 \( 1 + (-0.0627 + 0.998i)T^{2} \)
67 \( 1 + (1.06 - 1.67i)T + (-0.425 - 0.904i)T^{2} \)
71 \( 1 + (0.929 - 0.368i)T^{2} \)
73 \( 1 + (-1.03 - 1.24i)T + (-0.187 + 0.982i)T^{2} \)
79 \( 1 + (0.637 + 0.770i)T^{2} \)
83 \( 1 + (0.0800 + 1.27i)T + (-0.992 + 0.125i)T^{2} \)
89 \( 1 + (0.824 + 0.211i)T + (0.876 + 0.481i)T^{2} \)
97 \( 1 + (-0.110 - 1.74i)T + (-0.992 + 0.125i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.716684419974819778149688597293, −8.971045891038156032876908465758, −7.76274806992685525877133675106, −7.55882123109187170958669871688, −5.94332150296599301449656399441, −5.55027029876976317951033783175, −4.45404131220773972383351993407, −3.56578926132286398742521592254, −2.72187062550044097390484302765, −1.00011762086147921473333154331, 0.799574256329796414513515281778, 1.79803565395635955937090670251, 3.12218293610953102661843696924, 4.95883507128836731724128255428, 5.60807454260541907787259804671, 6.04119608253868964126231284890, 7.25743099208817198041600251652, 7.53540028034654109962637905189, 8.005398841777964622111997573421, 9.384228245024033125947791799823

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