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  + 2.50·2-s + 4.25·4-s + 5-s − 4.89·7-s + 5.63·8-s + 2.50·10-s − 5.13·11-s + 0.293·13-s − 12.2·14-s + 5.58·16-s − 4.78·17-s + 4.79·19-s + 4.25·20-s − 12.8·22-s − 5.18·23-s + 25-s + 0.734·26-s − 20.8·28-s − 8.87·29-s − 6.58·31-s + 2.69·32-s − 11.9·34-s − 4.89·35-s + 0.840·37-s + 11.9·38-s + 5.63·40-s + 2.51·41-s + ⋯
L(s)  = 1  + 1.76·2-s + 2.12·4-s + 0.447·5-s − 1.85·7-s + 1.99·8-s + 0.790·10-s − 1.54·11-s + 0.0814·13-s − 3.27·14-s + 1.39·16-s − 1.16·17-s + 1.09·19-s + 0.951·20-s − 2.73·22-s − 1.08·23-s + 0.200·25-s + 0.144·26-s − 3.93·28-s − 1.64·29-s − 1.18·31-s + 0.476·32-s − 2.05·34-s − 0.827·35-s + 0.138·37-s + 1.94·38-s + 0.890·40-s + 0.392·41-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 \( 1 - 2.50T + 2T^{2} \)
7 \( 1 + 4.89T + 7T^{2} \)
11 \( 1 + 5.13T + 11T^{2} \)
13 \( 1 - 0.293T + 13T^{2} \)
17 \( 1 + 4.78T + 17T^{2} \)
19 \( 1 - 4.79T + 19T^{2} \)
23 \( 1 + 5.18T + 23T^{2} \)
29 \( 1 + 8.87T + 29T^{2} \)
31 \( 1 + 6.58T + 31T^{2} \)
37 \( 1 - 0.840T + 37T^{2} \)
41 \( 1 - 2.51T + 41T^{2} \)
43 \( 1 - 9.39T + 43T^{2} \)
47 \( 1 + 8.06T + 47T^{2} \)
53 \( 1 - 11.4T + 53T^{2} \)
59 \( 1 + 1.04T + 59T^{2} \)
61 \( 1 - 6.77T + 61T^{2} \)
67 \( 1 + 14.0T + 67T^{2} \)
71 \( 1 + 2.08T + 71T^{2} \)
73 \( 1 - 6.37T + 73T^{2} \)
79 \( 1 - 0.566T + 79T^{2} \)
83 \( 1 + 6.12T + 83T^{2} \)
97 \( 1 + 3.68T + 97T^{2} \)
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

−7.57341485738314105293016898578, −7.14141453540294516122816935253, −6.25724083801928953921005851474, −5.73650817946596154250335579372, −5.26923413147789675021951000129, −4.15179138127968518965138624950, −3.49985222781124143855024423158, −2.72829893867766070638604148700, −2.13208635336916323987514522934, 0, 2.13208635336916323987514522934, 2.72829893867766070638604148700, 3.49985222781124143855024423158, 4.15179138127968518965138624950, 5.26923413147789675021951000129, 5.73650817946596154250335579372, 6.25724083801928953921005851474, 7.14141453540294516122816935253, 7.57341485738314105293016898578

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