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  − 0.938·2-s + 3-s − 1.11·4-s − 3.16·5-s − 0.938·6-s + 3.19·7-s + 2.92·8-s + 9-s + 2.96·10-s − 3.89·11-s − 1.11·12-s + 3.99·13-s − 2.99·14-s − 3.16·15-s − 0.508·16-s − 17-s − 0.938·18-s − 2.61·19-s + 3.53·20-s + 3.19·21-s + 3.65·22-s − 2.63·23-s + 2.92·24-s + 4.99·25-s − 3.75·26-s + 27-s − 3.57·28-s + ⋯
L(s)  = 1  − 0.663·2-s + 0.577·3-s − 0.559·4-s − 1.41·5-s − 0.383·6-s + 1.20·7-s + 1.03·8-s + 0.333·9-s + 0.938·10-s − 1.17·11-s − 0.323·12-s + 1.10·13-s − 0.800·14-s − 0.816·15-s − 0.127·16-s − 0.242·17-s − 0.221·18-s − 0.598·19-s + 0.791·20-s + 0.696·21-s + 0.779·22-s − 0.549·23-s + 0.597·24-s + 0.999·25-s − 0.735·26-s + 0.192·27-s − 0.675·28-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 \( 1 + 0.938T + 2T^{2} \)
5 \( 1 + 3.16T + 5T^{2} \)
7 \( 1 - 3.19T + 7T^{2} \)
11 \( 1 + 3.89T + 11T^{2} \)
13 \( 1 - 3.99T + 13T^{2} \)
19 \( 1 + 2.61T + 19T^{2} \)
23 \( 1 + 2.63T + 23T^{2} \)
29 \( 1 + 8.17T + 29T^{2} \)
31 \( 1 - 9.36T + 31T^{2} \)
37 \( 1 - 6.64T + 37T^{2} \)
41 \( 1 - 0.367T + 41T^{2} \)
43 \( 1 - 5.19T + 43T^{2} \)
47 \( 1 + 8.99T + 47T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 - 5.79T + 59T^{2} \)
61 \( 1 + 8.59T + 61T^{2} \)
67 \( 1 + 13.3T + 67T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
73 \( 1 + 8.17T + 73T^{2} \)
79 \( 1 - 6.91T + 79T^{2} \)
83 \( 1 - 4.30T + 83T^{2} \)
89 \( 1 + 0.653T + 89T^{2} \)
97 \( 1 - 17.7T + 97T^{2} \)
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

−7.84569239103401035386090949973, −7.36278721722553377612674167360, −6.15334961171482903482308107849, −5.18480200164560318616808084126, −4.33354656802469645052234438796, −4.17543276255365395886277994848, −3.16278593733041790314997297762, −2.07709805961306522737350870363, −1.09095786124540078117228624744, 0, 1.09095786124540078117228624744, 2.07709805961306522737350870363, 3.16278593733041790314997297762, 4.17543276255365395886277994848, 4.33354656802469645052234438796, 5.18480200164560318616808084126, 6.15334961171482903482308107849, 7.36278721722553377612674167360, 7.84569239103401035386090949973

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