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  + 0.983·2-s + 3-s − 1.03·4-s + 1.93·5-s + 0.983·6-s + 1.20·7-s − 2.98·8-s + 9-s + 1.90·10-s + 3.67·11-s − 1.03·12-s − 6.12·13-s + 1.18·14-s + 1.93·15-s − 0.865·16-s − 17-s + 0.983·18-s − 2.81·19-s − 1.99·20-s + 1.20·21-s + 3.61·22-s − 3.42·23-s − 2.98·24-s − 1.25·25-s − 6.02·26-s + 27-s − 1.24·28-s + ⋯
L(s)  = 1  + 0.695·2-s + 0.577·3-s − 0.516·4-s + 0.865·5-s + 0.401·6-s + 0.454·7-s − 1.05·8-s + 0.333·9-s + 0.601·10-s + 1.10·11-s − 0.298·12-s − 1.70·13-s + 0.315·14-s + 0.499·15-s − 0.216·16-s − 0.242·17-s + 0.231·18-s − 0.645·19-s − 0.447·20-s + 0.262·21-s + 0.770·22-s − 0.714·23-s − 0.608·24-s − 0.251·25-s − 1.18·26-s + 0.192·27-s − 0.234·28-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 \( 1 - 0.983T + 2T^{2} \)
5 \( 1 - 1.93T + 5T^{2} \)
7 \( 1 - 1.20T + 7T^{2} \)
11 \( 1 - 3.67T + 11T^{2} \)
13 \( 1 + 6.12T + 13T^{2} \)
19 \( 1 + 2.81T + 19T^{2} \)
23 \( 1 + 3.42T + 23T^{2} \)
29 \( 1 + 6.07T + 29T^{2} \)
31 \( 1 + 0.456T + 31T^{2} \)
37 \( 1 + 0.129T + 37T^{2} \)
41 \( 1 + 4.18T + 41T^{2} \)
43 \( 1 - 2.29T + 43T^{2} \)
47 \( 1 - 4.57T + 47T^{2} \)
53 \( 1 + 7.68T + 53T^{2} \)
59 \( 1 - 1.26T + 59T^{2} \)
61 \( 1 - 7.47T + 61T^{2} \)
67 \( 1 + 13.6T + 67T^{2} \)
71 \( 1 + 3.96T + 71T^{2} \)
73 \( 1 + 10.3T + 73T^{2} \)
79 \( 1 - 0.0902T + 79T^{2} \)
83 \( 1 + 9.40T + 83T^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 + 2.65T + 97T^{2} \)
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

−7.46715163184530083064279290446, −6.69485480826510565971006390314, −5.95210111274837331343418988981, −5.33961055918770452456920648827, −4.51452378645876974950876043867, −4.11187929735052618584622047534, −3.16370540206860146908485965207, −2.26968935123885978635825928333, −1.62328395581137710762333754652, 0, 1.62328395581137710762333754652, 2.26968935123885978635825928333, 3.16370540206860146908485965207, 4.11187929735052618584622047534, 4.51452378645876974950876043867, 5.33961055918770452456920648827, 5.95210111274837331343418988981, 6.69485480826510565971006390314, 7.46715163184530083064279290446

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