Properties

Degree 2
Conductor $ 2^{3} \cdot 251 $
Sign $-0.708 + 0.705i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.992 − 0.125i)2-s + (−1.58 − 0.119i)3-s + (0.968 + 0.248i)4-s + (1.55 + 0.317i)6-s + (−0.929 − 0.368i)8-s + (1.50 + 0.228i)9-s + (0.0174 − 1.38i)11-s + (−1.50 − 0.509i)12-s + (0.876 + 0.481i)16-s + (−0.417 + 0.455i)17-s + (−1.46 − 0.415i)18-s + (−0.180 − 0.832i)19-s + (−0.191 + 1.37i)22-s + (1.42 + 0.694i)24-s + (−0.425 + 0.904i)25-s + ⋯
L(s)  = 1  + (−0.992 − 0.125i)2-s + (−1.58 − 0.119i)3-s + (0.968 + 0.248i)4-s + (1.55 + 0.317i)6-s + (−0.929 − 0.368i)8-s + (1.50 + 0.228i)9-s + (0.0174 − 1.38i)11-s + (−1.50 − 0.509i)12-s + (0.876 + 0.481i)16-s + (−0.417 + 0.455i)17-s + (−1.46 − 0.415i)18-s + (−0.180 − 0.832i)19-s + (−0.191 + 1.37i)22-s + (1.42 + 0.694i)24-s + (−0.425 + 0.904i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.708 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.708 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2008\)    =    \(2^{3} \cdot 251\)
\( \varepsilon \)  =  $-0.708 + 0.705i$
motivic weight  =  \(0\)
character  :  $\chi_{2008} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2008,\ (\ :0),\ -0.708 + 0.705i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.2357170935\)
\(L(\frac12)\)  \(\approx\)  \(0.2357170935\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;251\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;251\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (0.992 + 0.125i)T \)
251 \( 1 + (-0.920 - 0.391i)T \)
good3 \( 1 + (1.58 + 0.119i)T + (0.988 + 0.150i)T^{2} \)
5 \( 1 + (0.425 - 0.904i)T^{2} \)
7 \( 1 + (-0.693 + 0.720i)T^{2} \)
11 \( 1 + (-0.0174 + 1.38i)T + (-0.999 - 0.0251i)T^{2} \)
13 \( 1 + (0.597 + 0.801i)T^{2} \)
17 \( 1 + (0.417 - 0.455i)T + (-0.0878 - 0.996i)T^{2} \)
19 \( 1 + (0.180 + 0.832i)T + (-0.910 + 0.414i)T^{2} \)
23 \( 1 + (0.974 + 0.224i)T^{2} \)
29 \( 1 + (0.379 - 0.925i)T^{2} \)
31 \( 1 + (-0.823 + 0.567i)T^{2} \)
37 \( 1 + (-0.0125 - 0.999i)T^{2} \)
41 \( 1 + (0.121 + 0.126i)T + (-0.0376 + 0.999i)T^{2} \)
43 \( 1 + (-0.811 + 0.559i)T + (0.356 - 0.934i)T^{2} \)
47 \( 1 + (-0.535 + 0.844i)T^{2} \)
53 \( 1 + (0.837 + 0.546i)T^{2} \)
59 \( 1 + (1.28 + 1.40i)T + (-0.0878 + 0.996i)T^{2} \)
61 \( 1 + (-0.260 + 0.965i)T^{2} \)
67 \( 1 + (0.936 - 1.75i)T + (-0.556 - 0.830i)T^{2} \)
71 \( 1 + (0.947 - 0.320i)T^{2} \)
73 \( 1 + (0.318 + 0.196i)T + (0.448 + 0.893i)T^{2} \)
79 \( 1 + (-0.762 + 0.647i)T^{2} \)
83 \( 1 + (0.472 + 0.0237i)T + (0.994 + 0.100i)T^{2} \)
89 \( 1 + (1.12 + 0.512i)T + (0.656 + 0.754i)T^{2} \)
97 \( 1 + (0.508 + 1.88i)T + (-0.863 + 0.503i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.024668240308706263039450560432, −8.404934484537437792097155458485, −7.35333944453931971729085938203, −6.71416077467544810188291101027, −5.92927808343044641909466175659, −5.45576287698250330381590818425, −4.15561115324907712493923295461, −2.96127975851140031674642127742, −1.52901603655967659721998698729, −0.31429394146850150168956161849, 1.31400256484655985124123036923, 2.47770092719892754502931898947, 4.17928674329835397725786415475, 4.97967968544832986929511148471, 5.94014580636499511811212846641, 6.45443748611725157614220854445, 7.27814230948234402724396563878, 7.891886675199280748618337687729, 9.081981926811418762980350115327, 9.753529303073684711906431024525

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