Properties

Label 2-8008-1.1-c1-0-17
Degree $2$
Conductor $8008$
Sign $1$
Analytic cond. $63.9442$
Root an. cond. $7.99651$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.29·3-s + 0.367·5-s + 7-s − 1.31·9-s − 11-s + 13-s − 0.478·15-s − 3.16·17-s − 2.94·19-s − 1.29·21-s − 2.33·23-s − 4.86·25-s + 5.60·27-s − 5.14·29-s − 5.54·31-s + 1.29·33-s + 0.367·35-s + 2.80·37-s − 1.29·39-s + 4.11·41-s + 3.28·43-s − 0.482·45-s − 1.06·47-s + 49-s + 4.11·51-s + 1.85·53-s − 0.367·55-s + ⋯
L(s)  = 1  − 0.750·3-s + 0.164·5-s + 0.377·7-s − 0.436·9-s − 0.301·11-s + 0.277·13-s − 0.123·15-s − 0.767·17-s − 0.676·19-s − 0.283·21-s − 0.486·23-s − 0.972·25-s + 1.07·27-s − 0.955·29-s − 0.995·31-s + 0.226·33-s + 0.0621·35-s + 0.461·37-s − 0.208·39-s + 0.642·41-s + 0.500·43-s − 0.0718·45-s − 0.154·47-s + 0.142·49-s + 0.576·51-s + 0.255·53-s − 0.0495·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8008\)    =    \(2^{3} \cdot 7 \cdot 11 \cdot 13\)
Sign: $1$
Analytic conductor: \(63.9442\)
Root analytic conductor: \(7.99651\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8008,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9332571107\)
\(L(\frac12)\) \(\approx\) \(0.9332571107\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 - T \)
good3 \( 1 + 1.29T + 3T^{2} \)
5 \( 1 - 0.367T + 5T^{2} \)
17 \( 1 + 3.16T + 17T^{2} \)
19 \( 1 + 2.94T + 19T^{2} \)
23 \( 1 + 2.33T + 23T^{2} \)
29 \( 1 + 5.14T + 29T^{2} \)
31 \( 1 + 5.54T + 31T^{2} \)
37 \( 1 - 2.80T + 37T^{2} \)
41 \( 1 - 4.11T + 41T^{2} \)
43 \( 1 - 3.28T + 43T^{2} \)
47 \( 1 + 1.06T + 47T^{2} \)
53 \( 1 - 1.85T + 53T^{2} \)
59 \( 1 - 3.95T + 59T^{2} \)
61 \( 1 + 3.98T + 61T^{2} \)
67 \( 1 + 3.64T + 67T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 - 6.64T + 79T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 - 9.60T + 89T^{2} \)
97 \( 1 + 16.6T + 97T^{2} \)
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

−7.84876629114027218962331458579, −7.08504475412674901858762251238, −6.26126932867098865745882456190, −5.79268446530268403688549482687, −5.18530809814256070662240247916, −4.34936944416241611272868141770, −3.66311080566121857659951962808, −2.49560348373047932845861779396, −1.81914571530957046968042464747, −0.48035809099157947999530886665, 0.48035809099157947999530886665, 1.81914571530957046968042464747, 2.49560348373047932845861779396, 3.66311080566121857659951962808, 4.34936944416241611272868141770, 5.18530809814256070662240247916, 5.79268446530268403688549482687, 6.26126932867098865745882456190, 7.08504475412674901858762251238, 7.84876629114027218962331458579

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