Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 23 \cdot 29 $
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 − 1.29·5-s + 2.07·7-s + 9-s − 3.34·11-s + 2.39·13-s − 1.29·15-s + 4.56·17-s − 2.47·19-s + 2.07·21-s + 23-s − 3.32·25-s + 27-s + 29-s − 10.5·31-s − 3.34·33-s − 2.68·35-s − 0.207·37-s + 2.39·39-s − 8.96·41-s − 7.63·43-s − 1.29·45-s − 0.402·47-s − 2.70·49-s + 4.56·51-s − 11.7·53-s + 4.33·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.579·5-s + 0.783·7-s + 0.333·9-s − 1.00·11-s + 0.664·13-s − 0.334·15-s + 1.10·17-s − 0.568·19-s + 0.452·21-s + 0.208·23-s − 0.664·25-s + 0.192·27-s + 0.185·29-s − 1.89·31-s − 0.582·33-s − 0.453·35-s − 0.0341·37-s + 0.383·39-s − 1.40·41-s − 1.16·43-s − 0.193·45-s − 0.0586·47-s − 0.386·49-s + 0.639·51-s − 1.60·53-s + 0.584·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8004\)    =    \(2^{2} \cdot 3 \cdot 23 \cdot 29\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8004} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8004,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;23,\;29\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;23,\;29\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 - T \)
good5 \( 1 + 1.29T + 5T^{2} \)
7 \( 1 - 2.07T + 7T^{2} \)
11 \( 1 + 3.34T + 11T^{2} \)
13 \( 1 - 2.39T + 13T^{2} \)
17 \( 1 - 4.56T + 17T^{2} \)
19 \( 1 + 2.47T + 19T^{2} \)
31 \( 1 + 10.5T + 31T^{2} \)
37 \( 1 + 0.207T + 37T^{2} \)
41 \( 1 + 8.96T + 41T^{2} \)
43 \( 1 + 7.63T + 43T^{2} \)
47 \( 1 + 0.402T + 47T^{2} \)
53 \( 1 + 11.7T + 53T^{2} \)
59 \( 1 - 3.18T + 59T^{2} \)
61 \( 1 + 3.69T + 61T^{2} \)
67 \( 1 - 2.52T + 67T^{2} \)
71 \( 1 - 8.77T + 71T^{2} \)
73 \( 1 - 2.09T + 73T^{2} \)
79 \( 1 + 2.54T + 79T^{2} \)
83 \( 1 - 4.26T + 83T^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 + 3.13T + 97T^{2} \)
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

−7.68835769474813816882526818220, −7.02430591709496586682066135471, −6.07671619225973810017825289848, −5.23447341534752719654548010819, −4.73698724256918459729658165974, −3.65283509783950124174739592151, −3.33751666564754310903987238677, −2.15919536834616792503931305142, −1.43558036318543306773175639961, 0, 1.43558036318543306773175639961, 2.15919536834616792503931305142, 3.33751666564754310903987238677, 3.65283509783950124174739592151, 4.73698724256918459729658165974, 5.23447341534752719654548010819, 6.07671619225973810017825289848, 7.02430591709496586682066135471, 7.68835769474813816882526818220

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