Properties

Degree 2
Conductor $ 3^{2} \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  + 1.93·2-s + 1.73·4-s + 2.82·5-s − 3.97·7-s − 0.514·8-s + 5.45·10-s + 4.70·11-s − 5.71·13-s − 7.67·14-s − 4.46·16-s − 5.11·17-s + 0.224·19-s + 4.89·20-s + 9.08·22-s + 23-s + 2.95·25-s − 11.0·26-s − 6.88·28-s + 29-s + 5.16·31-s − 7.59·32-s − 9.87·34-s − 11.2·35-s − 5.77·37-s + 0.434·38-s − 1.45·40-s + 0.469·41-s + ⋯
L(s)  = 1  + 1.36·2-s + 0.866·4-s + 1.26·5-s − 1.50·7-s − 0.181·8-s + 1.72·10-s + 1.41·11-s − 1.58·13-s − 2.05·14-s − 1.11·16-s − 1.23·17-s + 0.0515·19-s + 1.09·20-s + 1.93·22-s + 0.208·23-s + 0.591·25-s − 2.16·26-s − 1.30·28-s + 0.185·29-s + 0.926·31-s − 1.34·32-s − 1.69·34-s − 1.89·35-s − 0.949·37-s + 0.0704·38-s − 0.229·40-s + 0.0733·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6003} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6003,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;23,\;29\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;23,\;29\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
23 \( 1 - T \)
29 \( 1 - T \)
good2 \( 1 - 1.93T + 2T^{2} \)
5 \( 1 - 2.82T + 5T^{2} \)
7 \( 1 + 3.97T + 7T^{2} \)
11 \( 1 - 4.70T + 11T^{2} \)
13 \( 1 + 5.71T + 13T^{2} \)
17 \( 1 + 5.11T + 17T^{2} \)
19 \( 1 - 0.224T + 19T^{2} \)
31 \( 1 - 5.16T + 31T^{2} \)
37 \( 1 + 5.77T + 37T^{2} \)
41 \( 1 - 0.469T + 41T^{2} \)
43 \( 1 - 2.68T + 43T^{2} \)
47 \( 1 + 12.2T + 47T^{2} \)
53 \( 1 - 3.63T + 53T^{2} \)
59 \( 1 + 8.78T + 59T^{2} \)
61 \( 1 - 4.68T + 61T^{2} \)
67 \( 1 + 15.9T + 67T^{2} \)
71 \( 1 + 10.0T + 71T^{2} \)
73 \( 1 - 8.41T + 73T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 + 11.6T + 83T^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 + 5.53T + 97T^{2} \)
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

−7.13745598953974412763643089506, −6.71308095714389386666661753246, −6.22503201644019236411068418460, −5.65347899716522760065387667922, −4.73691185068728763853151939585, −4.21250808592068490020251734117, −3.17237132393420759670655331823, −2.66904240348122054480694534880, −1.75227669797089786275276216124, 0, 1.75227669797089786275276216124, 2.66904240348122054480694534880, 3.17237132393420759670655331823, 4.21250808592068490020251734117, 4.73691185068728763853151939585, 5.65347899716522760065387667922, 6.22503201644019236411068418460, 6.71308095714389386666661753246, 7.13745598953974412763643089506

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