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 + 3.10·3-s + 4-s + 1.73·5-s + 3.10·6-s + 3.71·7-s + 8-s + 6.61·9-s + 1.73·10-s + 0.351·11-s + 3.10·12-s − 3.27·13-s + 3.71·14-s + 5.37·15-s + 16-s + 0.752·17-s + 6.61·18-s − 19-s + 1.73·20-s + 11.5·21-s + 0.351·22-s − 6.92·23-s + 3.10·24-s − 1.98·25-s − 3.27·26-s + 11.1·27-s + 3.71·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.78·3-s + 0.5·4-s + 0.776·5-s + 1.26·6-s + 1.40·7-s + 0.353·8-s + 2.20·9-s + 0.548·10-s + 0.105·11-s + 0.894·12-s − 0.909·13-s + 0.993·14-s + 1.38·15-s + 0.250·16-s + 0.182·17-s + 1.55·18-s − 0.229·19-s + 0.388·20-s + 2.51·21-s + 0.0748·22-s − 1.44·23-s + 0.632·24-s − 0.397·25-s − 0.643·26-s + 2.15·27-s + 0.702·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $8.930346515$
$L(\frac12)$  $\approx$  $8.930346515$
$L(\frac{3}{2})$   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 \( 1 - 3.10T + 3T^{2} \)
5 \( 1 - 1.73T + 5T^{2} \)
7 \( 1 - 3.71T + 7T^{2} \)
11 \( 1 - 0.351T + 11T^{2} \)
13 \( 1 + 3.27T + 13T^{2} \)
17 \( 1 - 0.752T + 17T^{2} \)
23 \( 1 + 6.92T + 23T^{2} \)
29 \( 1 - 1.99T + 29T^{2} \)
31 \( 1 + 1.76T + 31T^{2} \)
37 \( 1 + 9.46T + 37T^{2} \)
41 \( 1 - 7.34T + 41T^{2} \)
43 \( 1 - 4.84T + 43T^{2} \)
47 \( 1 - 9.66T + 47T^{2} \)
53 \( 1 + 12.6T + 53T^{2} \)
59 \( 1 + 2.58T + 59T^{2} \)
61 \( 1 + 3.16T + 61T^{2} \)
67 \( 1 - 14.5T + 67T^{2} \)
71 \( 1 - 2.53T + 71T^{2} \)
73 \( 1 + 4.14T + 73T^{2} \)
79 \( 1 - 2.86T + 79T^{2} \)
83 \( 1 - 13.5T + 83T^{2} \)
89 \( 1 - 13.1T + 89T^{2} \)
97 \( 1 - 1.69T + 97T^{2} \)
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

−7.84996096998405135753605297474, −7.40663936610516581720528412856, −6.49721171332429395224742117880, −5.59377296034726052123100839563, −4.85827599097144185838667517281, −4.19307864444928110051427074550, −3.54959528652984940189038288690, −2.44395838147375659093461683932, −2.14450465079399996780562938026, −1.43562708965236668339140880515, 1.43562708965236668339140880515, 2.14450465079399996780562938026, 2.44395838147375659093461683932, 3.54959528652984940189038288690, 4.19307864444928110051427074550, 4.85827599097144185838667517281, 5.59377296034726052123100839563, 6.49721171332429395224742117880, 7.40663936610516581720528412856, 7.84996096998405135753605297474

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