Properties

Label 2-2008-1.1-c1-0-34
Degree $2$
Conductor $2008$
Sign $1$
Analytic cond. $16.0339$
Root an. cond. $4.00424$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.51·3-s − 0.472·5-s + 4.16·7-s + 3.33·9-s − 0.304·11-s + 3.93·13-s − 1.18·15-s + 5.23·17-s − 3.05·19-s + 10.4·21-s + 7.82·23-s − 4.77·25-s + 0.847·27-s − 7.79·29-s − 9.12·31-s − 0.766·33-s − 1.96·35-s − 7.45·37-s + 9.90·39-s − 5.32·41-s + 10.2·43-s − 1.57·45-s + 8.20·47-s + 10.3·49-s + 13.1·51-s − 8.29·53-s + 0.143·55-s + ⋯
L(s)  = 1  + 1.45·3-s − 0.211·5-s + 1.57·7-s + 1.11·9-s − 0.0918·11-s + 1.09·13-s − 0.306·15-s + 1.26·17-s − 0.701·19-s + 2.28·21-s + 1.63·23-s − 0.955·25-s + 0.163·27-s − 1.44·29-s − 1.63·31-s − 0.133·33-s − 0.332·35-s − 1.22·37-s + 1.58·39-s − 0.832·41-s + 1.56·43-s − 0.234·45-s + 1.19·47-s + 1.48·49-s + 1.84·51-s − 1.13·53-s + 0.0193·55-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2008\)    =    \(2^{3} \cdot 251\)
Sign: $1$
Analytic conductor: \(16.0339\)
Root analytic conductor: \(4.00424\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2008,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.556432660\)
\(L(\frac12)\) \(\approx\) \(3.556432660\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
251 \( 1 + T \)
good3 \( 1 - 2.51T + 3T^{2} \)
5 \( 1 + 0.472T + 5T^{2} \)
7 \( 1 - 4.16T + 7T^{2} \)
11 \( 1 + 0.304T + 11T^{2} \)
13 \( 1 - 3.93T + 13T^{2} \)
17 \( 1 - 5.23T + 17T^{2} \)
19 \( 1 + 3.05T + 19T^{2} \)
23 \( 1 - 7.82T + 23T^{2} \)
29 \( 1 + 7.79T + 29T^{2} \)
31 \( 1 + 9.12T + 31T^{2} \)
37 \( 1 + 7.45T + 37T^{2} \)
41 \( 1 + 5.32T + 41T^{2} \)
43 \( 1 - 10.2T + 43T^{2} \)
47 \( 1 - 8.20T + 47T^{2} \)
53 \( 1 + 8.29T + 53T^{2} \)
59 \( 1 - 3.03T + 59T^{2} \)
61 \( 1 + 0.845T + 61T^{2} \)
67 \( 1 + 5.94T + 67T^{2} \)
71 \( 1 - 6.20T + 71T^{2} \)
73 \( 1 - 10.1T + 73T^{2} \)
79 \( 1 + 9.74T + 79T^{2} \)
83 \( 1 + 4.98T + 83T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 - 0.396T + 97T^{2} \)
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

−8.817379041078856144775583208564, −8.539235587563849885741158603275, −7.54816456889488908190454564336, −7.42645578504342275940130719342, −5.83420110743480903754409063346, −5.06340094504036134278444406881, −3.91497878372608141029367459923, −3.42414511290684254817427013670, −2.14073974637787167363712516092, −1.39955481773998976069350032439, 1.39955481773998976069350032439, 2.14073974637787167363712516092, 3.42414511290684254817427013670, 3.91497878372608141029367459923, 5.06340094504036134278444406881, 5.83420110743480903754409063346, 7.42645578504342275940130719342, 7.54816456889488908190454564336, 8.539235587563849885741158603275, 8.817379041078856144775583208564

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