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 + 0.821·3-s + 4-s + 3.37·5-s + 0.821·6-s − 5.13·7-s + 8-s − 2.32·9-s + 3.37·10-s − 1.78·11-s + 0.821·12-s + 4.39·13-s − 5.13·14-s + 2.76·15-s + 16-s + 6.07·17-s − 2.32·18-s − 19-s + 3.37·20-s − 4.21·21-s − 1.78·22-s − 4.63·23-s + 0.821·24-s + 6.35·25-s + 4.39·26-s − 4.37·27-s − 5.13·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.474·3-s + 0.5·4-s + 1.50·5-s + 0.335·6-s − 1.94·7-s + 0.353·8-s − 0.775·9-s + 1.06·10-s − 0.537·11-s + 0.237·12-s + 1.22·13-s − 1.37·14-s + 0.714·15-s + 0.250·16-s + 1.47·17-s − 0.548·18-s − 0.229·19-s + 0.753·20-s − 0.920·21-s − 0.379·22-s − 0.966·23-s + 0.167·24-s + 1.27·25-s + 0.862·26-s − 0.841·27-s − 0.970·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.198685423$
$L(\frac12)$  $\approx$  $4.198685423$
$L(\frac{3}{2})$   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 \( 1 - 0.821T + 3T^{2} \)
5 \( 1 - 3.37T + 5T^{2} \)
7 \( 1 + 5.13T + 7T^{2} \)
11 \( 1 + 1.78T + 11T^{2} \)
13 \( 1 - 4.39T + 13T^{2} \)
17 \( 1 - 6.07T + 17T^{2} \)
23 \( 1 + 4.63T + 23T^{2} \)
29 \( 1 - 7.00T + 29T^{2} \)
31 \( 1 - 2.01T + 31T^{2} \)
37 \( 1 + 6.47T + 37T^{2} \)
41 \( 1 - 11.3T + 41T^{2} \)
43 \( 1 - 6.64T + 43T^{2} \)
47 \( 1 - 2.22T + 47T^{2} \)
53 \( 1 + 2.56T + 53T^{2} \)
59 \( 1 - 8.88T + 59T^{2} \)
61 \( 1 + 14.2T + 61T^{2} \)
67 \( 1 + 0.602T + 67T^{2} \)
71 \( 1 - 9.73T + 71T^{2} \)
73 \( 1 - 16.3T + 73T^{2} \)
79 \( 1 + 1.18T + 79T^{2} \)
83 \( 1 - 6.51T + 83T^{2} \)
89 \( 1 + 9.24T + 89T^{2} \)
97 \( 1 + 10.0T + 97T^{2} \)
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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

−7.84125282851199917871777953029, −6.79838000230911144067795700113, −6.18923294027308170647139533575, −5.84616513093960988100536911672, −5.40179361998222920246692251609, −4.06632611904578302107805536367, −3.31079956103105761247406805850, −2.83052927445341251085633273647, −2.16405832248826948370427081571, −0.877426701942868224593131785966, 0.877426701942868224593131785966, 2.16405832248826948370427081571, 2.83052927445341251085633273647, 3.31079956103105761247406805850, 4.06632611904578302107805536367, 5.40179361998222920246692251609, 5.84616513093960988100536911672, 6.18923294027308170647139533575, 6.79838000230911144067795700113, 7.84125282851199917871777953029

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