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  + 0.901·2-s − 3-s − 1.18·4-s − 2.82·5-s − 0.901·6-s − 0.878·7-s − 2.87·8-s + 9-s − 2.54·10-s + 4.81·11-s + 1.18·12-s − 5.14·13-s − 0.792·14-s + 2.82·15-s − 0.216·16-s − 17-s + 0.901·18-s − 0.823·19-s + 3.35·20-s + 0.878·21-s + 4.34·22-s + 2.70·23-s + 2.87·24-s + 2.98·25-s − 4.63·26-s − 27-s + 1.04·28-s + ⋯
L(s)  = 1  + 0.637·2-s − 0.577·3-s − 0.593·4-s − 1.26·5-s − 0.368·6-s − 0.332·7-s − 1.01·8-s + 0.333·9-s − 0.805·10-s + 1.45·11-s + 0.342·12-s − 1.42·13-s − 0.211·14-s + 0.729·15-s − 0.0541·16-s − 0.242·17-s + 0.212·18-s − 0.189·19-s + 0.749·20-s + 0.191·21-s + 0.925·22-s + 0.564·23-s + 0.586·24-s + 0.596·25-s − 0.909·26-s − 0.192·27-s + 0.197·28-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 \( 1 - 0.901T + 2T^{2} \)
5 \( 1 + 2.82T + 5T^{2} \)
7 \( 1 + 0.878T + 7T^{2} \)
11 \( 1 - 4.81T + 11T^{2} \)
13 \( 1 + 5.14T + 13T^{2} \)
19 \( 1 + 0.823T + 19T^{2} \)
23 \( 1 - 2.70T + 23T^{2} \)
29 \( 1 + 7.58T + 29T^{2} \)
31 \( 1 - 6.60T + 31T^{2} \)
37 \( 1 - 7.06T + 37T^{2} \)
41 \( 1 - 2.65T + 41T^{2} \)
43 \( 1 + 2.47T + 43T^{2} \)
47 \( 1 - 6.04T + 47T^{2} \)
53 \( 1 + 2.88T + 53T^{2} \)
59 \( 1 - 4.30T + 59T^{2} \)
61 \( 1 - 12.7T + 61T^{2} \)
67 \( 1 - 5.27T + 67T^{2} \)
71 \( 1 - 0.559T + 71T^{2} \)
73 \( 1 - 8.67T + 73T^{2} \)
79 \( 1 + 11.9T + 79T^{2} \)
83 \( 1 - 6.37T + 83T^{2} \)
89 \( 1 + 0.915T + 89T^{2} \)
97 \( 1 + 0.734T + 97T^{2} \)
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

−7.34308749350712298507161099778, −6.74238852303051539182742014802, −6.07724036037831842174191070366, −5.20212579141039996013808567883, −4.55197205428835466763710565813, −4.03687965058738087326630565792, −3.48784949069710902993137718037, −2.45806444107566643858309961794, −0.914030171459538142267267857182, 0, 0.914030171459538142267267857182, 2.45806444107566643858309961794, 3.48784949069710902993137718037, 4.03687965058738087326630565792, 4.55197205428835466763710565813, 5.20212579141039996013808567883, 6.07724036037831842174191070366, 6.74238852303051539182742014802, 7.34308749350712298507161099778

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