Properties

Label 2-2008-1.1-c1-0-12
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  − 3.19·3-s − 2.92·5-s + 4.80·7-s + 7.22·9-s + 5.38·11-s + 3.67·13-s + 9.36·15-s − 1.72·17-s − 7.21·19-s − 15.3·21-s + 4.61·23-s + 3.56·25-s − 13.5·27-s + 8.38·29-s − 2.29·31-s − 17.2·33-s − 14.0·35-s − 1.44·37-s − 11.7·39-s + 2.85·41-s − 8.84·43-s − 21.1·45-s + 5.70·47-s + 16.0·49-s + 5.51·51-s + 10.4·53-s − 15.7·55-s + ⋯
L(s)  = 1  − 1.84·3-s − 1.30·5-s + 1.81·7-s + 2.40·9-s + 1.62·11-s + 1.01·13-s + 2.41·15-s − 0.418·17-s − 1.65·19-s − 3.34·21-s + 0.961·23-s + 0.713·25-s − 2.60·27-s + 1.55·29-s − 0.412·31-s − 3.00·33-s − 2.37·35-s − 0.237·37-s − 1.88·39-s + 0.446·41-s − 1.34·43-s − 3.15·45-s + 0.831·47-s + 2.29·49-s + 0.772·51-s + 1.43·53-s − 2.12·55-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

\(L(1)\) \(\approx\) \(1.020755677\)
\(L(\frac12)\) \(\approx\) \(1.020755677\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.19T + 3T^{2} \)
5 \( 1 + 2.92T + 5T^{2} \)
7 \( 1 - 4.80T + 7T^{2} \)
11 \( 1 - 5.38T + 11T^{2} \)
13 \( 1 - 3.67T + 13T^{2} \)
17 \( 1 + 1.72T + 17T^{2} \)
19 \( 1 + 7.21T + 19T^{2} \)
23 \( 1 - 4.61T + 23T^{2} \)
29 \( 1 - 8.38T + 29T^{2} \)
31 \( 1 + 2.29T + 31T^{2} \)
37 \( 1 + 1.44T + 37T^{2} \)
41 \( 1 - 2.85T + 41T^{2} \)
43 \( 1 + 8.84T + 43T^{2} \)
47 \( 1 - 5.70T + 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 + 8.60T + 59T^{2} \)
61 \( 1 - 4.17T + 61T^{2} \)
67 \( 1 + 13.8T + 67T^{2} \)
71 \( 1 + 2.80T + 71T^{2} \)
73 \( 1 - 5.31T + 73T^{2} \)
79 \( 1 - 2.35T + 79T^{2} \)
83 \( 1 - 9.40T + 83T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 - 3.06T + 97T^{2} \)
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

−8.862660066450586823871345813818, −8.431712699783113590799445311692, −7.40051302640634203242391283308, −6.70320796383373037208082444009, −6.06044774390059697747273476257, −4.94227805510728250296479012057, −4.37965037888227003045861476523, −3.91060588494395982928321754005, −1.64377408864306479243444151947, −0.816095557398412414611607501294, 0.816095557398412414611607501294, 1.64377408864306479243444151947, 3.91060588494395982928321754005, 4.37965037888227003045861476523, 4.94227805510728250296479012057, 6.06044774390059697747273476257, 6.70320796383373037208082444009, 7.40051302640634203242391283308, 8.431712699783113590799445311692, 8.862660066450586823871345813818

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