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 − 0.764·3-s + 4-s − 0.445·5-s − 0.764·6-s − 0.938·7-s + 8-s − 2.41·9-s − 0.445·10-s + 2.33·11-s − 0.764·12-s − 6.49·13-s − 0.938·14-s + 0.340·15-s + 16-s + 1.93·17-s − 2.41·18-s − 19-s − 0.445·20-s + 0.717·21-s + 2.33·22-s + 3.50·23-s − 0.764·24-s − 4.80·25-s − 6.49·26-s + 4.13·27-s − 0.938·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.441·3-s + 0.5·4-s − 0.199·5-s − 0.311·6-s − 0.354·7-s + 0.353·8-s − 0.805·9-s − 0.140·10-s + 0.705·11-s − 0.220·12-s − 1.80·13-s − 0.250·14-s + 0.0879·15-s + 0.250·16-s + 0.470·17-s − 0.569·18-s − 0.229·19-s − 0.0996·20-s + 0.156·21-s + 0.498·22-s + 0.730·23-s − 0.155·24-s − 0.960·25-s − 1.27·26-s + 0.796·27-s − 0.177·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.782217746$
$L(\frac12)$  $\approx$  $1.782217746$
$L(\frac{3}{2})$   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 \( 1 + 0.764T + 3T^{2} \)
5 \( 1 + 0.445T + 5T^{2} \)
7 \( 1 + 0.938T + 7T^{2} \)
11 \( 1 - 2.33T + 11T^{2} \)
13 \( 1 + 6.49T + 13T^{2} \)
17 \( 1 - 1.93T + 17T^{2} \)
23 \( 1 - 3.50T + 23T^{2} \)
29 \( 1 + 1.31T + 29T^{2} \)
31 \( 1 - 4.83T + 31T^{2} \)
37 \( 1 - 2.27T + 37T^{2} \)
41 \( 1 + 8.03T + 41T^{2} \)
43 \( 1 + 2.86T + 43T^{2} \)
47 \( 1 - 2.15T + 47T^{2} \)
53 \( 1 - 3.08T + 53T^{2} \)
59 \( 1 - 1.71T + 59T^{2} \)
61 \( 1 - 2.46T + 61T^{2} \)
67 \( 1 - 4.61T + 67T^{2} \)
71 \( 1 - 8.01T + 71T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 - 0.375T + 79T^{2} \)
83 \( 1 - 6.21T + 83T^{2} \)
89 \( 1 - 14.0T + 89T^{2} \)
97 \( 1 + 16.8T + 97T^{2} \)
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

−7.76163885780367253700751630940, −6.80606493365228872997493935137, −6.56220572553182516583737499023, −5.56466952073145280160419670315, −5.13089758714433051990516891196, −4.39279828599621922641242124761, −3.51190347696440592495138546847, −2.81588366492008312100704428287, −1.98347729837650387447684636666, −0.58003162821411874244693176842, 0.58003162821411874244693176842, 1.98347729837650387447684636666, 2.81588366492008312100704428287, 3.51190347696440592495138546847, 4.39279828599621922641242124761, 5.13089758714433051990516891196, 5.56466952073145280160419670315, 6.56220572553182516583737499023, 6.80606493365228872997493935137, 7.76163885780367253700751630940

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