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 + 0.677·3-s + 4-s − 4.37·5-s + 0.677·6-s − 1.03·7-s + 8-s − 2.54·9-s − 4.37·10-s + 1.08·11-s + 0.677·12-s + 1.73·13-s − 1.03·14-s − 2.96·15-s + 16-s + 3.23·17-s − 2.54·18-s + 19-s − 4.37·20-s − 0.704·21-s + 1.08·22-s + 0.112·23-s + 0.677·24-s + 14.1·25-s + 1.73·26-s − 3.75·27-s − 1.03·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.391·3-s + 0.5·4-s − 1.95·5-s + 0.276·6-s − 0.392·7-s + 0.353·8-s − 0.846·9-s − 1.38·10-s + 0.328·11-s + 0.195·12-s + 0.482·13-s − 0.277·14-s − 0.765·15-s + 0.250·16-s + 0.784·17-s − 0.598·18-s + 0.229·19-s − 0.978·20-s − 0.153·21-s + 0.231·22-s + 0.0235·23-s + 0.138·24-s + 2.83·25-s + 0.340·26-s − 0.722·27-s − 0.196·28-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 \( 1 - 0.677T + 3T^{2} \)
5 \( 1 + 4.37T + 5T^{2} \)
7 \( 1 + 1.03T + 7T^{2} \)
11 \( 1 - 1.08T + 11T^{2} \)
13 \( 1 - 1.73T + 13T^{2} \)
17 \( 1 - 3.23T + 17T^{2} \)
23 \( 1 - 0.112T + 23T^{2} \)
29 \( 1 - 3.19T + 29T^{2} \)
31 \( 1 + 0.880T + 31T^{2} \)
37 \( 1 + 5.39T + 37T^{2} \)
41 \( 1 + 3.58T + 41T^{2} \)
43 \( 1 - 4.67T + 43T^{2} \)
47 \( 1 - 11.3T + 47T^{2} \)
53 \( 1 - 6.66T + 53T^{2} \)
59 \( 1 + 6.07T + 59T^{2} \)
61 \( 1 + 3.81T + 61T^{2} \)
67 \( 1 + 11.5T + 67T^{2} \)
71 \( 1 + 8.52T + 71T^{2} \)
73 \( 1 + 3.18T + 73T^{2} \)
79 \( 1 + 9.57T + 79T^{2} \)
83 \( 1 - 12.4T + 83T^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 - 9.17T + 97T^{2} \)
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

−7.50181936031565776886311098587, −6.91341378576502551509006828483, −6.06744457883328529459972282034, −5.31986245545725517173868199290, −4.42485292890336333099899626793, −3.82090724970939958830200364347, −3.25181406709781208803291481005, −2.74655705810819930538885782170, −1.20991165283800632370276430635, 0, 1.20991165283800632370276430635, 2.74655705810819930538885782170, 3.25181406709781208803291481005, 3.82090724970939958830200364347, 4.42485292890336333099899626793, 5.31986245545725517173868199290, 6.06744457883328529459972282034, 6.91341378576502551509006828483, 7.50181936031565776886311098587

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