Properties

Degree 2
Conductor $ 2^{3} \cdot 251 $
Sign $-0.448 + 0.893i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.187 − 0.982i)2-s + (−0.948 + 1.67i)3-s + (−0.929 + 0.368i)4-s + (1.82 + 0.617i)6-s + (0.535 + 0.844i)8-s + (−1.39 − 2.31i)9-s + (−0.513 + 0.0913i)11-s + (0.265 − 1.90i)12-s + (0.728 − 0.684i)16-s + (0.356 − 0.504i)17-s + (−2.01 + 1.79i)18-s + (−1.96 + 0.298i)19-s + (0.185 + 0.487i)22-s + (−1.92 + 0.0966i)24-s + (−0.992 − 0.125i)25-s + ⋯
L(s)  = 1  + (−0.187 − 0.982i)2-s + (−0.948 + 1.67i)3-s + (−0.929 + 0.368i)4-s + (1.82 + 0.617i)6-s + (0.535 + 0.844i)8-s + (−1.39 − 2.31i)9-s + (−0.513 + 0.0913i)11-s + (0.265 − 1.90i)12-s + (0.728 − 0.684i)16-s + (0.356 − 0.504i)17-s + (−2.01 + 1.79i)18-s + (−1.96 + 0.298i)19-s + (0.185 + 0.487i)22-s + (−1.92 + 0.0966i)24-s + (−0.992 − 0.125i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2008\)    =    \(2^{3} \cdot 251\)
\( \varepsilon \)  =  $-0.448 + 0.893i$
motivic weight  =  \(0\)
character  :  $\chi_{2008} (75, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2008,\ (\ :0),\ -0.448 + 0.893i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.2218724236\)
\(L(\frac12)\)  \(\approx\)  \(0.2218724236\)
\(L(1)\)   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 \( 1 + (0.187 + 0.982i)T \)
251 \( 1 + (-0.793 - 0.607i)T \)
good3 \( 1 + (0.948 - 1.67i)T + (-0.514 - 0.857i)T^{2} \)
5 \( 1 + (0.992 + 0.125i)T^{2} \)
7 \( 1 + (-0.260 + 0.965i)T^{2} \)
11 \( 1 + (0.513 - 0.0913i)T + (0.938 - 0.344i)T^{2} \)
13 \( 1 + (-0.899 + 0.437i)T^{2} \)
17 \( 1 + (-0.356 + 0.504i)T + (-0.332 - 0.942i)T^{2} \)
19 \( 1 + (1.96 - 0.298i)T + (0.954 - 0.297i)T^{2} \)
23 \( 1 + (0.999 - 0.0251i)T^{2} \)
29 \( 1 + (0.675 + 0.737i)T^{2} \)
31 \( 1 + (0.556 - 0.830i)T^{2} \)
37 \( 1 + (0.984 + 0.175i)T^{2} \)
41 \( 1 + (0.173 + 0.642i)T + (-0.863 + 0.503i)T^{2} \)
43 \( 1 + (-0.665 + 0.993i)T + (-0.379 - 0.925i)T^{2} \)
47 \( 1 + (-0.0627 - 0.998i)T^{2} \)
53 \( 1 + (0.236 + 0.971i)T^{2} \)
59 \( 1 + (0.543 + 0.768i)T + (-0.332 + 0.942i)T^{2} \)
61 \( 1 + (-0.850 + 0.525i)T^{2} \)
67 \( 1 + (0.272 - 0.177i)T + (0.402 - 0.915i)T^{2} \)
71 \( 1 + (0.137 + 0.990i)T^{2} \)
73 \( 1 + (-0.197 + 1.74i)T + (-0.974 - 0.224i)T^{2} \)
79 \( 1 + (0.910 + 0.414i)T^{2} \)
83 \( 1 + (-1.49 + 1.26i)T + (0.162 - 0.986i)T^{2} \)
89 \( 1 + (-1.90 - 0.591i)T + (0.823 + 0.567i)T^{2} \)
97 \( 1 + (1.70 + 1.05i)T + (0.448 + 0.893i)T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.313237334929846132942688497127, −8.783067778792524654267420055945, −7.83663437431085362085096329724, −6.41090816631417779954197067901, −5.54042708581932766822745215437, −4.85704303324201795597382669748, −4.09850327244563875833235481622, −3.47869670381705612016115514040, −2.24807703436125587173447181175, −0.20370052962473972530254328727, 1.28918908493841281558432125950, 2.44415422216245220157870439583, 4.22360477545343462901018131261, 5.20495495083296886166310312492, 5.99106942017985857561449218599, 6.40039418915888384978715154318, 7.16183401086037433115922658361, 8.013472663262279277521424206083, 8.226320109038991319530771662691, 9.356671047625382138799593437027

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