Properties

Degree 2
Conductor $ 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3.21·3-s − 2.32·5-s − 7-s + 7.33·9-s − 11-s − 13-s + 7.46·15-s + 7.61·17-s + 7.80·19-s + 3.21·21-s − 9.32·23-s + 0.393·25-s − 13.9·27-s + 3.30·29-s − 9.29·31-s + 3.21·33-s + 2.32·35-s + 3.92·37-s + 3.21·39-s − 10.6·41-s + 7.32·43-s − 17.0·45-s − 10.8·47-s + 49-s − 24.4·51-s − 0.263·53-s + 2.32·55-s + ⋯
L(s)  = 1  − 1.85·3-s − 1.03·5-s − 0.377·7-s + 2.44·9-s − 0.301·11-s − 0.277·13-s + 1.92·15-s + 1.84·17-s + 1.78·19-s + 0.701·21-s − 1.94·23-s + 0.0786·25-s − 2.68·27-s + 0.613·29-s − 1.66·31-s + 0.559·33-s + 0.392·35-s + 0.645·37-s + 0.514·39-s − 1.66·41-s + 1.11·43-s − 2.53·45-s − 1.58·47-s + 0.142·49-s − 3.42·51-s − 0.0361·53-s + 0.313·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4004} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4573750648$
$L(\frac12)$  $\approx$  $0.4573750648$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;11,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;11,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
7 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 + T \)
good3 \( 1 + 3.21T + 3T^{2} \)
5 \( 1 + 2.32T + 5T^{2} \)
17 \( 1 - 7.61T + 17T^{2} \)
19 \( 1 - 7.80T + 19T^{2} \)
23 \( 1 + 9.32T + 23T^{2} \)
29 \( 1 - 3.30T + 29T^{2} \)
31 \( 1 + 9.29T + 31T^{2} \)
37 \( 1 - 3.92T + 37T^{2} \)
41 \( 1 + 10.6T + 41T^{2} \)
43 \( 1 - 7.32T + 43T^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 + 0.263T + 53T^{2} \)
59 \( 1 + 7.99T + 59T^{2} \)
61 \( 1 + 0.0316T + 61T^{2} \)
67 \( 1 - 9.12T + 67T^{2} \)
71 \( 1 - 3.34T + 71T^{2} \)
73 \( 1 + 14.3T + 73T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 + 6.71T + 83T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + 7.56T + 97T^{2} \)
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

−8.015173136827951313840841588020, −7.63871295992185270023506113984, −7.00018673221243394588473219297, −6.04012355619565932836352551464, −5.52104687075762202723765042230, −4.90725683031360722367497098872, −3.92509800379475493951588395028, −3.29375380042648363337391567031, −1.53047492106899188794709707974, −0.44653826345841452861847187969, 0.44653826345841452861847187969, 1.53047492106899188794709707974, 3.29375380042648363337391567031, 3.92509800379475493951588395028, 4.90725683031360722367497098872, 5.52104687075762202723765042230, 6.04012355619565932836352551464, 7.00018673221243394588473219297, 7.63871295992185270023506113984, 8.015173136827951313840841588020

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