Properties

Degree 2
Conductor $ 3 \cdot 17 \cdot 157 $
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  − 0.251·2-s + 3-s − 1.93·4-s + 2.63·5-s − 0.251·6-s + 1.22·7-s + 0.988·8-s + 9-s − 0.662·10-s − 0.402·11-s − 1.93·12-s − 0.940·13-s − 0.306·14-s + 2.63·15-s + 3.62·16-s − 17-s − 0.251·18-s + 1.25·19-s − 5.10·20-s + 1.22·21-s + 0.101·22-s − 8.86·23-s + 0.988·24-s + 1.94·25-s + 0.236·26-s + 27-s − 2.36·28-s + ⋯
L(s)  = 1  − 0.177·2-s + 0.577·3-s − 0.968·4-s + 1.17·5-s − 0.102·6-s + 0.461·7-s + 0.349·8-s + 0.333·9-s − 0.209·10-s − 0.121·11-s − 0.559·12-s − 0.260·13-s − 0.0819·14-s + 0.680·15-s + 0.906·16-s − 0.242·17-s − 0.0591·18-s + 0.288·19-s − 1.14·20-s + 0.266·21-s + 0.0215·22-s − 1.84·23-s + 0.201·24-s + 0.389·25-s + 0.0463·26-s + 0.192·27-s − 0.446·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8007\)    =    \(3 \cdot 17 \cdot 157\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8007} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8007,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;17,\;157\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;17,\;157\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
17 \( 1 + T \)
157 \( 1 - T \)
good2 \( 1 + 0.251T + 2T^{2} \)
5 \( 1 - 2.63T + 5T^{2} \)
7 \( 1 - 1.22T + 7T^{2} \)
11 \( 1 + 0.402T + 11T^{2} \)
13 \( 1 + 0.940T + 13T^{2} \)
19 \( 1 - 1.25T + 19T^{2} \)
23 \( 1 + 8.86T + 23T^{2} \)
29 \( 1 - 1.22T + 29T^{2} \)
31 \( 1 + 6.62T + 31T^{2} \)
37 \( 1 - 2.87T + 37T^{2} \)
41 \( 1 + 5.03T + 41T^{2} \)
43 \( 1 + 7.04T + 43T^{2} \)
47 \( 1 + 2.20T + 47T^{2} \)
53 \( 1 + 4.59T + 53T^{2} \)
59 \( 1 + 6.65T + 59T^{2} \)
61 \( 1 - 4.50T + 61T^{2} \)
67 \( 1 + 0.159T + 67T^{2} \)
71 \( 1 + 6.42T + 71T^{2} \)
73 \( 1 - 3.73T + 73T^{2} \)
79 \( 1 - 6.14T + 79T^{2} \)
83 \( 1 + 14.4T + 83T^{2} \)
89 \( 1 + 0.974T + 89T^{2} \)
97 \( 1 + 10.7T + 97T^{2} \)
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

−7.78886990102954564855829152673, −6.82803037081648086423658745853, −5.97680927520449513324877553408, −5.37182981844869122324269543028, −4.68932857747846213748168715847, −3.95200585602432033145297165370, −3.08748392396160959199929151468, −2.00476714670813446305752586081, −1.51701475169946186890728211837, 0, 1.51701475169946186890728211837, 2.00476714670813446305752586081, 3.08748392396160959199929151468, 3.95200585602432033145297165370, 4.68932857747846213748168715847, 5.37182981844869122324269543028, 5.97680927520449513324877553408, 6.82803037081648086423658745853, 7.78886990102954564855829152673

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