↑

Properties

Label 2-2008-2008.1419-c0-0-0
Degree $2$
Conductor $2008$
Sign $-0.448 - 0.893i$
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  + (−0.187 + 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

Degree: \(2\)
Conductor: \(2008\)    =    \(2^{3} \cdot 251\)
Sign: $-0.448 - 0.893i$
Analytic conductor: \(1.00212\)
Root analytic conductor: \(1.00106\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2008} (1419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2008,\ (\ :0),\ -0.448 - 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2218724236\)
\(L(\frac12)\) \(\approx\) \(0.2218724236\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.187 - 0.982i)T \)
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
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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