Properties

Degree 2
Conductor $ 2 \cdot 19 \cdot 211 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2.19·3-s + 4-s + 1.33·5-s − 2.19·6-s − 4.99·7-s + 8-s + 1.80·9-s + 1.33·10-s + 3.23·11-s − 2.19·12-s + 3.93·13-s − 4.99·14-s − 2.92·15-s + 16-s + 2.30·17-s + 1.80·18-s − 19-s + 1.33·20-s + 10.9·21-s + 3.23·22-s + 2.41·23-s − 2.19·24-s − 3.21·25-s + 3.93·26-s + 2.61·27-s − 4.99·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.26·3-s + 0.5·4-s + 0.597·5-s − 0.895·6-s − 1.88·7-s + 0.353·8-s + 0.602·9-s + 0.422·10-s + 0.975·11-s − 0.632·12-s + 1.09·13-s − 1.33·14-s − 0.756·15-s + 0.250·16-s + 0.558·17-s + 0.425·18-s − 0.229·19-s + 0.298·20-s + 2.38·21-s + 0.689·22-s + 0.502·23-s − 0.447·24-s − 0.642·25-s + 0.771·26-s + 0.503·27-s − 0.943·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.812656462$
$L(\frac12)$  $\approx$  $1.812656462$
$L(\frac{3}{2})$   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 \( 1 + 2.19T + 3T^{2} \)
5 \( 1 - 1.33T + 5T^{2} \)
7 \( 1 + 4.99T + 7T^{2} \)
11 \( 1 - 3.23T + 11T^{2} \)
13 \( 1 - 3.93T + 13T^{2} \)
17 \( 1 - 2.30T + 17T^{2} \)
23 \( 1 - 2.41T + 23T^{2} \)
29 \( 1 - 7.68T + 29T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 - 3.67T + 37T^{2} \)
41 \( 1 + 8.72T + 41T^{2} \)
43 \( 1 - 9.15T + 43T^{2} \)
47 \( 1 + 9.14T + 47T^{2} \)
53 \( 1 - 5.22T + 53T^{2} \)
59 \( 1 + 9.07T + 59T^{2} \)
61 \( 1 + 1.01T + 61T^{2} \)
67 \( 1 + 14.2T + 67T^{2} \)
71 \( 1 + 5.03T + 71T^{2} \)
73 \( 1 - 0.0673T + 73T^{2} \)
79 \( 1 - 17.2T + 79T^{2} \)
83 \( 1 + 7.67T + 83T^{2} \)
89 \( 1 - 16.3T + 89T^{2} \)
97 \( 1 + 11.3T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.40508793697920087687709248885, −6.69938228394360345283333986676, −6.12274884212901313523283617236, −6.06945275064089812303289136464, −5.26765096268114038069325078021, −4.30734974150123213386978938900, −3.52835124052068841632880658088, −2.98639311447330555940431298118, −1.67021689388451605050734148316, −0.64892563264492005995570478631, 0.64892563264492005995570478631, 1.67021689388451605050734148316, 2.98639311447330555940431298118, 3.52835124052068841632880658088, 4.30734974150123213386978938900, 5.26765096268114038069325078021, 6.06945275064089812303289136464, 6.12274884212901313523283617236, 6.69938228394360345283333986676, 7.40508793697920087687709248885

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