Properties

Degree 2
Conductor $ 3^{2} \cdot 5 \cdot 89 $
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  − 2.23·2-s + 2.98·4-s + 5-s + 1.87·7-s − 2.20·8-s − 2.23·10-s − 2.02·11-s + 5.20·13-s − 4.17·14-s − 1.05·16-s + 1.73·17-s − 5.29·19-s + 2.98·20-s + 4.53·22-s − 3.33·23-s + 25-s − 11.6·26-s + 5.58·28-s + 3.53·29-s − 4.98·31-s + 6.75·32-s − 3.88·34-s + 1.87·35-s − 7.17·37-s + 11.8·38-s − 2.20·40-s + 1.08·41-s + ⋯
L(s)  = 1  − 1.57·2-s + 1.49·4-s + 0.447·5-s + 0.706·7-s − 0.779·8-s − 0.706·10-s − 0.611·11-s + 1.44·13-s − 1.11·14-s − 0.262·16-s + 0.421·17-s − 1.21·19-s + 0.667·20-s + 0.965·22-s − 0.695·23-s + 0.200·25-s − 2.27·26-s + 1.05·28-s + 0.657·29-s − 0.895·31-s + 1.19·32-s − 0.666·34-s + 0.316·35-s − 1.17·37-s + 1.91·38-s − 0.348·40-s + 0.169·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4005\)    =    \(3^{2} \cdot 5 \cdot 89\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4005} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4005,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;89\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;89\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
5 \( 1 - T \)
89 \( 1 + T \)
good2 \( 1 + 2.23T + 2T^{2} \)
7 \( 1 - 1.87T + 7T^{2} \)
11 \( 1 + 2.02T + 11T^{2} \)
13 \( 1 - 5.20T + 13T^{2} \)
17 \( 1 - 1.73T + 17T^{2} \)
19 \( 1 + 5.29T + 19T^{2} \)
23 \( 1 + 3.33T + 23T^{2} \)
29 \( 1 - 3.53T + 29T^{2} \)
31 \( 1 + 4.98T + 31T^{2} \)
37 \( 1 + 7.17T + 37T^{2} \)
41 \( 1 - 1.08T + 41T^{2} \)
43 \( 1 + 4.73T + 43T^{2} \)
47 \( 1 + 7.45T + 47T^{2} \)
53 \( 1 + 12.1T + 53T^{2} \)
59 \( 1 + 3.66T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 + 9.52T + 67T^{2} \)
71 \( 1 + 2.81T + 71T^{2} \)
73 \( 1 - 1.11T + 73T^{2} \)
79 \( 1 + 6.37T + 79T^{2} \)
83 \( 1 + 11.6T + 83T^{2} \)
97 \( 1 - 16.0T + 97T^{2} \)
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

−8.399013250017887316809026624357, −7.68726710243763894327397606154, −6.75030951538665387578134252615, −6.15383888112426620499133653209, −5.23584889385749985958688611012, −4.26785720196981901954513981987, −3.10989391739364177464801798496, −1.91200569855489123789239403615, −1.42994015789105176831538915534, 0, 1.42994015789105176831538915534, 1.91200569855489123789239403615, 3.10989391739364177464801798496, 4.26785720196981901954513981987, 5.23584889385749985958688611012, 6.15383888112426620499133653209, 6.75030951538665387578134252615, 7.68726710243763894327397606154, 8.399013250017887316809026624357

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