Properties

Degree 2
Conductor $ 2 \cdot 19 \cdot 211 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 1.39·3-s + 4-s + 0.539·5-s − 1.39·6-s + 5.05·7-s + 8-s − 1.04·9-s + 0.539·10-s − 1.12·11-s − 1.39·12-s − 2.68·13-s + 5.05·14-s − 0.754·15-s + 16-s − 4.30·17-s − 1.04·18-s − 19-s + 0.539·20-s − 7.07·21-s − 1.12·22-s + 7.92·23-s − 1.39·24-s − 4.70·25-s − 2.68·26-s + 5.65·27-s + 5.05·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.807·3-s + 0.5·4-s + 0.241·5-s − 0.570·6-s + 1.91·7-s + 0.353·8-s − 0.348·9-s + 0.170·10-s − 0.339·11-s − 0.403·12-s − 0.745·13-s + 1.35·14-s − 0.194·15-s + 0.250·16-s − 1.04·17-s − 0.246·18-s − 0.229·19-s + 0.120·20-s − 1.54·21-s − 0.239·22-s + 1.65·23-s − 0.285·24-s − 0.941·25-s − 0.527·26-s + 1.08·27-s + 0.955·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8018\)    =    \(2 \cdot 19 \cdot 211\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8018} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.823964691$
$L(\frac12)$  $\approx$  $2.823964691$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;19,\;211\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;19,\;211\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
19 \( 1 + T \)
211 \( 1 + T \)
good3 \( 1 + 1.39T + 3T^{2} \)
5 \( 1 - 0.539T + 5T^{2} \)
7 \( 1 - 5.05T + 7T^{2} \)
11 \( 1 + 1.12T + 11T^{2} \)
13 \( 1 + 2.68T + 13T^{2} \)
17 \( 1 + 4.30T + 17T^{2} \)
23 \( 1 - 7.92T + 23T^{2} \)
29 \( 1 + 0.281T + 29T^{2} \)
31 \( 1 + 6.56T + 31T^{2} \)
37 \( 1 - 1.71T + 37T^{2} \)
41 \( 1 - 0.408T + 41T^{2} \)
43 \( 1 - 5.34T + 43T^{2} \)
47 \( 1 + 3.58T + 47T^{2} \)
53 \( 1 + 3.58T + 53T^{2} \)
59 \( 1 - 8.94T + 59T^{2} \)
61 \( 1 - 13.1T + 61T^{2} \)
67 \( 1 - 7.49T + 67T^{2} \)
71 \( 1 - 9.98T + 71T^{2} \)
73 \( 1 + 7.18T + 73T^{2} \)
79 \( 1 - 16.4T + 79T^{2} \)
83 \( 1 - 7.00T + 83T^{2} \)
89 \( 1 + 4.37T + 89T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
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

−7.65468525712461011933210344143, −7.09288022714840386305559300881, −6.29261642965882708706946518963, −5.45450789885037841515749746018, −5.07052497081745451418224641841, −4.65310426706034694113225316781, −3.72275668046139438395917820629, −2.44654422246039743139480355444, −1.99773434100919927613276242010, −0.77451743182509831186676039861, 0.77451743182509831186676039861, 1.99773434100919927613276242010, 2.44654422246039743139480355444, 3.72275668046139438395917820629, 4.65310426706034694113225316781, 5.07052497081745451418224641841, 5.45450789885037841515749746018, 6.29261642965882708706946518963, 7.09288022714840386305559300881, 7.65468525712461011933210344143

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