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 − 3.15·3-s + 4-s − 2.25·5-s + 3.15·6-s − 1.49·7-s − 8-s + 6.93·9-s + 2.25·10-s − 3.03·11-s − 3.15·12-s + 3.87·13-s + 1.49·14-s + 7.12·15-s + 16-s − 3.17·17-s − 6.93·18-s − 19-s − 2.25·20-s + 4.72·21-s + 3.03·22-s + 3.24·23-s + 3.15·24-s + 0.104·25-s − 3.87·26-s − 12.4·27-s − 1.49·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.82·3-s + 0.5·4-s − 1.01·5-s + 1.28·6-s − 0.566·7-s − 0.353·8-s + 2.31·9-s + 0.714·10-s − 0.915·11-s − 0.910·12-s + 1.07·13-s + 0.400·14-s + 1.83·15-s + 0.250·16-s − 0.770·17-s − 1.63·18-s − 0.229·19-s − 0.505·20-s + 1.03·21-s + 0.647·22-s + 0.677·23-s + 0.643·24-s + 0.0208·25-s − 0.760·26-s − 2.39·27-s − 0.283·28-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 \( 1 + 3.15T + 3T^{2} \)
5 \( 1 + 2.25T + 5T^{2} \)
7 \( 1 + 1.49T + 7T^{2} \)
11 \( 1 + 3.03T + 11T^{2} \)
13 \( 1 - 3.87T + 13T^{2} \)
17 \( 1 + 3.17T + 17T^{2} \)
23 \( 1 - 3.24T + 23T^{2} \)
29 \( 1 - 1.46T + 29T^{2} \)
31 \( 1 + 10.7T + 31T^{2} \)
37 \( 1 - 3.23T + 37T^{2} \)
41 \( 1 - 3.81T + 41T^{2} \)
43 \( 1 - 1.14T + 43T^{2} \)
47 \( 1 + 0.559T + 47T^{2} \)
53 \( 1 + 1.40T + 53T^{2} \)
59 \( 1 + 0.460T + 59T^{2} \)
61 \( 1 + 2.40T + 61T^{2} \)
67 \( 1 + 9.34T + 67T^{2} \)
71 \( 1 - 4.71T + 71T^{2} \)
73 \( 1 + 8.63T + 73T^{2} \)
79 \( 1 - 6.74T + 79T^{2} \)
83 \( 1 - 2.80T + 83T^{2} \)
89 \( 1 - 16.9T + 89T^{2} \)
97 \( 1 - 7.51T + 97T^{2} \)
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

−7.44806494108182201706306296577, −6.75987960322119459167175225813, −6.18450442654322462846088293658, −5.58700579556002372845747855812, −4.76765172450369432455438998046, −4.02828203079071836014154630039, −3.20067008122439609623417446637, −1.85918416288452683196556759977, −0.71541135313625066094150948922, 0, 0.71541135313625066094150948922, 1.85918416288452683196556759977, 3.20067008122439609623417446637, 4.02828203079071836014154630039, 4.76765172450369432455438998046, 5.58700579556002372845747855812, 6.18450442654322462846088293658, 6.75987960322119459167175225813, 7.44806494108182201706306296577

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