Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 167 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 4.17·5-s − 1.78·7-s + 9-s + 4.35·11-s + 2.54·13-s − 4.17·15-s + 0.0716·17-s − 7.16·19-s − 1.78·21-s − 2.75·23-s + 12.3·25-s + 27-s + 5.44·29-s + 3.34·31-s + 4.35·33-s + 7.45·35-s − 6.04·37-s + 2.54·39-s + 3.33·41-s + 3.47·43-s − 4.17·45-s + 1.84·47-s − 3.80·49-s + 0.0716·51-s − 10.4·53-s − 18.1·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.86·5-s − 0.675·7-s + 0.333·9-s + 1.31·11-s + 0.707·13-s − 1.07·15-s + 0.0173·17-s − 1.64·19-s − 0.389·21-s − 0.573·23-s + 2.47·25-s + 0.192·27-s + 1.01·29-s + 0.600·31-s + 0.757·33-s + 1.25·35-s − 0.993·37-s + 0.408·39-s + 0.520·41-s + 0.530·43-s − 0.621·45-s + 0.269·47-s − 0.544·49-s + 0.0100·51-s − 1.43·53-s − 2.44·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4008} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4008,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;167\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;167\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 - T \)
167 \( 1 + T \)
good5 \( 1 + 4.17T + 5T^{2} \)
7 \( 1 + 1.78T + 7T^{2} \)
11 \( 1 - 4.35T + 11T^{2} \)
13 \( 1 - 2.54T + 13T^{2} \)
17 \( 1 - 0.0716T + 17T^{2} \)
19 \( 1 + 7.16T + 19T^{2} \)
23 \( 1 + 2.75T + 23T^{2} \)
29 \( 1 - 5.44T + 29T^{2} \)
31 \( 1 - 3.34T + 31T^{2} \)
37 \( 1 + 6.04T + 37T^{2} \)
41 \( 1 - 3.33T + 41T^{2} \)
43 \( 1 - 3.47T + 43T^{2} \)
47 \( 1 - 1.84T + 47T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 - 7.84T + 59T^{2} \)
61 \( 1 + 6.59T + 61T^{2} \)
67 \( 1 - 3.06T + 67T^{2} \)
71 \( 1 + 9.41T + 71T^{2} \)
73 \( 1 + 8.85T + 73T^{2} \)
79 \( 1 - 0.0583T + 79T^{2} \)
83 \( 1 - 0.584T + 83T^{2} \)
89 \( 1 + 3.61T + 89T^{2} \)
97 \( 1 - 0.334T + 97T^{2} \)
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

−8.261308399523202473016275915726, −7.42030982010376576729402480997, −6.63690024444596914365212973951, −6.23177791322386184595316847202, −4.64591863974464679050618781892, −4.05728668501461928240188977553, −3.61130929562089539633045162198, −2.74681037654593591062045816033, −1.30034461778575977840305754336, 0, 1.30034461778575977840305754336, 2.74681037654593591062045816033, 3.61130929562089539633045162198, 4.05728668501461928240188977553, 4.64591863974464679050618781892, 6.23177791322386184595316847202, 6.63690024444596914365212973951, 7.42030982010376576729402480997, 8.261308399523202473016275915726

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