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 + 2.54·5-s − 4.05·7-s + 9-s + 0.833·11-s + 2.46·13-s + 2.54·15-s − 1.83·17-s − 5.92·19-s − 4.05·21-s − 3.92·23-s + 1.46·25-s + 27-s − 7.78·29-s − 8.46·31-s + 0.833·33-s − 10.3·35-s + 3.96·37-s + 2.46·39-s − 11.0·41-s − 10.1·43-s + 2.54·45-s + 0.498·47-s + 9.40·49-s − 1.83·51-s + 2.45·53-s + 2.12·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.13·5-s − 1.53·7-s + 0.333·9-s + 0.251·11-s + 0.682·13-s + 0.656·15-s − 0.446·17-s − 1.35·19-s − 0.883·21-s − 0.817·23-s + 0.293·25-s + 0.192·27-s − 1.44·29-s − 1.51·31-s + 0.145·33-s − 1.74·35-s + 0.651·37-s + 0.394·39-s − 1.72·41-s − 1.54·43-s + 0.379·45-s + 0.0726·47-s + 1.34·49-s − 0.257·51-s + 0.337·53-s + 0.285·55-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 \( 1 - 2.54T + 5T^{2} \)
7 \( 1 + 4.05T + 7T^{2} \)
11 \( 1 - 0.833T + 11T^{2} \)
13 \( 1 - 2.46T + 13T^{2} \)
17 \( 1 + 1.83T + 17T^{2} \)
19 \( 1 + 5.92T + 19T^{2} \)
23 \( 1 + 3.92T + 23T^{2} \)
29 \( 1 + 7.78T + 29T^{2} \)
31 \( 1 + 8.46T + 31T^{2} \)
37 \( 1 - 3.96T + 37T^{2} \)
41 \( 1 + 11.0T + 41T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 - 0.498T + 47T^{2} \)
53 \( 1 - 2.45T + 53T^{2} \)
59 \( 1 - 7.33T + 59T^{2} \)
61 \( 1 - 2.76T + 61T^{2} \)
67 \( 1 - 10.9T + 67T^{2} \)
71 \( 1 + 3.97T + 71T^{2} \)
73 \( 1 - 6.83T + 73T^{2} \)
79 \( 1 - 7.34T + 79T^{2} \)
83 \( 1 + 4.66T + 83T^{2} \)
89 \( 1 + 4.56T + 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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

−8.333390893076308603186033526008, −7.15610823911444660490330537122, −6.51503020773670896102596896349, −6.06094850303353407652506154091, −5.25095186554722852427379767092, −3.89823007376272264226664012646, −3.52790058170199856889482901447, −2.33776850107007713865957999060, −1.75956024153102106506725367570, 0, 1.75956024153102106506725367570, 2.33776850107007713865957999060, 3.52790058170199856889482901447, 3.89823007376272264226664012646, 5.25095186554722852427379767092, 6.06094850303353407652506154091, 6.51503020773670896102596896349, 7.15610823911444660490330537122, 8.333390893076308603186033526008

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