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  − 2.52·2-s − 3-s + 4.35·4-s − 1.87·5-s + 2.52·6-s − 3.72·7-s − 5.92·8-s + 9-s + 4.72·10-s − 2.94·11-s − 4.35·12-s − 4.80·13-s + 9.37·14-s + 1.87·15-s + 6.24·16-s − 17-s − 2.52·18-s − 2.96·19-s − 8.15·20-s + 3.72·21-s + 7.42·22-s − 3.55·23-s + 5.92·24-s − 1.48·25-s + 12.1·26-s − 27-s − 16.1·28-s + ⋯
L(s)  = 1  − 1.78·2-s − 0.577·3-s + 2.17·4-s − 0.838·5-s + 1.02·6-s − 1.40·7-s − 2.09·8-s + 0.333·9-s + 1.49·10-s − 0.888·11-s − 1.25·12-s − 1.33·13-s + 2.50·14-s + 0.484·15-s + 1.56·16-s − 0.242·17-s − 0.594·18-s − 0.680·19-s − 1.82·20-s + 0.812·21-s + 1.58·22-s − 0.741·23-s + 1.21·24-s − 0.297·25-s + 2.37·26-s − 0.192·27-s − 3.06·28-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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
17 \( 1 + T \)
157 \( 1 + T \)
good2 \( 1 + 2.52T + 2T^{2} \)
5 \( 1 + 1.87T + 5T^{2} \)
7 \( 1 + 3.72T + 7T^{2} \)
11 \( 1 + 2.94T + 11T^{2} \)
13 \( 1 + 4.80T + 13T^{2} \)
19 \( 1 + 2.96T + 19T^{2} \)
23 \( 1 + 3.55T + 23T^{2} \)
29 \( 1 + 2.12T + 29T^{2} \)
31 \( 1 + 1.66T + 31T^{2} \)
37 \( 1 - 0.184T + 37T^{2} \)
41 \( 1 - 12.0T + 41T^{2} \)
43 \( 1 - 0.542T + 43T^{2} \)
47 \( 1 - 8.00T + 47T^{2} \)
53 \( 1 - 5.77T + 53T^{2} \)
59 \( 1 + 4.76T + 59T^{2} \)
61 \( 1 + 8.87T + 61T^{2} \)
67 \( 1 - 6.37T + 67T^{2} \)
71 \( 1 - 0.652T + 71T^{2} \)
73 \( 1 + 10.6T + 73T^{2} \)
79 \( 1 - 5.92T + 79T^{2} \)
83 \( 1 - 7.36T + 83T^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 + 6.53T + 97T^{2} \)
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

−7.37290030943568439309086552855, −7.30847135544416957859637341782, −6.28386658870456411459072078525, −5.83446821729116962819690531606, −4.65747435267863403285727271843, −3.76074503444398907518467392778, −2.70632639687168502225349538745, −2.13152250082447210827293216964, −0.57305346918020445966960042984, 0, 0.57305346918020445966960042984, 2.13152250082447210827293216964, 2.70632639687168502225349538745, 3.76074503444398907518467392778, 4.65747435267863403285727271843, 5.83446821729116962819690531606, 6.28386658870456411459072078525, 7.30847135544416957859637341782, 7.37290030943568439309086552855

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