Properties

Degree 2
Conductor $ 2 \cdot 19 \cdot 211 $
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-s − 3.17·3-s + 4-s + 4.15·5-s + 3.17·6-s + 1.45·7-s − 8-s + 7.05·9-s − 4.15·10-s − 3.03·11-s − 3.17·12-s + 3.91·13-s − 1.45·14-s − 13.1·15-s + 16-s − 0.0576·17-s − 7.05·18-s − 19-s + 4.15·20-s − 4.61·21-s + 3.03·22-s + 0.939·23-s + 3.17·24-s + 12.2·25-s − 3.91·26-s − 12.8·27-s + 1.45·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.83·3-s + 0.5·4-s + 1.85·5-s + 1.29·6-s + 0.549·7-s − 0.353·8-s + 2.35·9-s − 1.31·10-s − 0.915·11-s − 0.915·12-s + 1.08·13-s − 0.388·14-s − 3.40·15-s + 0.250·16-s − 0.0139·17-s − 1.66·18-s − 0.229·19-s + 0.928·20-s − 1.00·21-s + 0.647·22-s + 0.195·23-s + 0.647·24-s + 2.45·25-s − 0.768·26-s − 2.47·27-s + 0.274·28-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  =  1
Selberg data  =  $(2,\ 8018,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;19,\;211\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]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 \( 1 + 3.17T + 3T^{2} \)
5 \( 1 - 4.15T + 5T^{2} \)
7 \( 1 - 1.45T + 7T^{2} \)
11 \( 1 + 3.03T + 11T^{2} \)
13 \( 1 - 3.91T + 13T^{2} \)
17 \( 1 + 0.0576T + 17T^{2} \)
23 \( 1 - 0.939T + 23T^{2} \)
29 \( 1 + 9.74T + 29T^{2} \)
31 \( 1 - 3.98T + 31T^{2} \)
37 \( 1 + 10.1T + 37T^{2} \)
41 \( 1 - 6.40T + 41T^{2} \)
43 \( 1 + 0.156T + 43T^{2} \)
47 \( 1 + 3.44T + 47T^{2} \)
53 \( 1 - 8.72T + 53T^{2} \)
59 \( 1 + 6.77T + 59T^{2} \)
61 \( 1 - 0.287T + 61T^{2} \)
67 \( 1 + 0.311T + 67T^{2} \)
71 \( 1 + 4.31T + 71T^{2} \)
73 \( 1 + 12.8T + 73T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 + 3.11T + 83T^{2} \)
89 \( 1 + 9.56T + 89T^{2} \)
97 \( 1 + 18.1T + 97T^{2} \)
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

−7.21374876413022442649173587804, −6.69468181329075984050384325833, −5.94162338882312357235358614981, −5.59492126551759083668911195746, −5.15139240014627772693820006618, −4.17992800226292469045082397053, −2.76133667342133814621749843580, −1.65515114123767730542187621573, −1.34669111682136299649716726448, 0, 1.34669111682136299649716726448, 1.65515114123767730542187621573, 2.76133667342133814621749843580, 4.17992800226292469045082397053, 5.15139240014627772693820006618, 5.59492126551759083668911195746, 5.94162338882312357235358614981, 6.69468181329075984050384325833, 7.21374876413022442649173587804

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