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 + 0.836·3-s + 4-s − 3.34·5-s + 0.836·6-s − 0.767·7-s + 8-s − 2.30·9-s − 3.34·10-s − 6.48·11-s + 0.836·12-s + 1.07·13-s − 0.767·14-s − 2.79·15-s + 16-s − 0.0698·17-s − 2.30·18-s − 19-s − 3.34·20-s − 0.641·21-s − 6.48·22-s + 6.21·23-s + 0.836·24-s + 6.18·25-s + 1.07·26-s − 4.43·27-s − 0.767·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.482·3-s + 0.5·4-s − 1.49·5-s + 0.341·6-s − 0.290·7-s + 0.353·8-s − 0.766·9-s − 1.05·10-s − 1.95·11-s + 0.241·12-s + 0.297·13-s − 0.205·14-s − 0.722·15-s + 0.250·16-s − 0.0169·17-s − 0.542·18-s − 0.229·19-s − 0.747·20-s − 0.140·21-s − 1.38·22-s + 1.29·23-s + 0.170·24-s + 1.23·25-s + 0.210·26-s − 0.853·27-s − 0.145·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.409429775$
$L(\frac12)$  $\approx$  $1.409429775$
$L(\frac{3}{2})$   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 \( 1 - 0.836T + 3T^{2} \)
5 \( 1 + 3.34T + 5T^{2} \)
7 \( 1 + 0.767T + 7T^{2} \)
11 \( 1 + 6.48T + 11T^{2} \)
13 \( 1 - 1.07T + 13T^{2} \)
17 \( 1 + 0.0698T + 17T^{2} \)
23 \( 1 - 6.21T + 23T^{2} \)
29 \( 1 + 5.02T + 29T^{2} \)
31 \( 1 + 4.55T + 31T^{2} \)
37 \( 1 - 9.11T + 37T^{2} \)
41 \( 1 + 11.1T + 41T^{2} \)
43 \( 1 - 9.28T + 43T^{2} \)
47 \( 1 + 3.90T + 47T^{2} \)
53 \( 1 - 1.97T + 53T^{2} \)
59 \( 1 + 5.39T + 59T^{2} \)
61 \( 1 - 9.57T + 61T^{2} \)
67 \( 1 - 8.23T + 67T^{2} \)
71 \( 1 - 6.56T + 71T^{2} \)
73 \( 1 - 1.31T + 73T^{2} \)
79 \( 1 - 9.27T + 79T^{2} \)
83 \( 1 + 7.18T + 83T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + 9.54T + 97T^{2} \)
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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

−7.85359794844351966057500934445, −7.30586656402759400543166659184, −6.48988863142033206237082292930, −5.47884043340788939555738728949, −5.08514686803978713835771333083, −4.16303089354402680246124436196, −3.40910588745114651051002197033, −2.97170953974756755634805812397, −2.18029572193767702689849035549, −0.47855091485073462279581707985, 0.47855091485073462279581707985, 2.18029572193767702689849035549, 2.97170953974756755634805812397, 3.40910588745114651051002197033, 4.16303089354402680246124436196, 5.08514686803978713835771333083, 5.47884043340788939555738728949, 6.48988863142033206237082292930, 7.30586656402759400543166659184, 7.85359794844351966057500934445

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