Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 1.01·3-s + 4-s − 1.68·5-s + 1.01·6-s + 2.57·7-s + 8-s − 1.97·9-s − 1.68·10-s + 3.74·11-s + 1.01·12-s − 1.44·13-s + 2.57·14-s − 1.70·15-s + 16-s − 5.57·17-s − 1.97·18-s + 19-s − 1.68·20-s + 2.60·21-s + 3.74·22-s − 4.47·23-s + 1.01·24-s − 2.15·25-s − 1.44·26-s − 5.03·27-s + 2.57·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.583·3-s + 0.5·4-s − 0.753·5-s + 0.412·6-s + 0.974·7-s + 0.353·8-s − 0.659·9-s − 0.532·10-s + 1.12·11-s + 0.291·12-s − 0.400·13-s + 0.688·14-s − 0.439·15-s + 0.250·16-s − 1.35·17-s − 0.466·18-s + 0.229·19-s − 0.376·20-s + 0.568·21-s + 0.798·22-s − 0.932·23-s + 0.206·24-s − 0.431·25-s − 0.283·26-s − 0.968·27-s + 0.487·28-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 \( 1 - 1.01T + 3T^{2} \)
5 \( 1 + 1.68T + 5T^{2} \)
7 \( 1 - 2.57T + 7T^{2} \)
11 \( 1 - 3.74T + 11T^{2} \)
13 \( 1 + 1.44T + 13T^{2} \)
17 \( 1 + 5.57T + 17T^{2} \)
23 \( 1 + 4.47T + 23T^{2} \)
29 \( 1 - 1.04T + 29T^{2} \)
31 \( 1 + 1.57T + 31T^{2} \)
37 \( 1 + 7.99T + 37T^{2} \)
41 \( 1 + 9.16T + 41T^{2} \)
43 \( 1 + 6.28T + 43T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 + 13.6T + 53T^{2} \)
59 \( 1 + 6.06T + 59T^{2} \)
61 \( 1 + 5.39T + 61T^{2} \)
67 \( 1 + 9.88T + 67T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 - 12.8T + 73T^{2} \)
79 \( 1 - 0.509T + 79T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 - 7.25T + 89T^{2} \)
97 \( 1 - 8.92T + 97T^{2} \)
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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

−7.64017410811256444387321766969, −6.71234779554564495019101332566, −6.19147535355571727641624653177, −5.13580161133522489376605041500, −4.63986736973905698533583983557, −3.80575319581583324799960351183, −3.37941722819673826351898803498, −2.22642088984907930691821598557, −1.66478504063893895381239873255, 0, 1.66478504063893895381239873255, 2.22642088984907930691821598557, 3.37941722819673826351898803498, 3.80575319581583324799960351183, 4.63986736973905698533583983557, 5.13580161133522489376605041500, 6.19147535355571727641624653177, 6.71234779554564495019101332566, 7.64017410811256444387321766969

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