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 − 1.03·3-s + 4-s + 0.118·5-s − 1.03·6-s + 2.13·7-s + 8-s − 1.92·9-s + 0.118·10-s + 5.30·11-s − 1.03·12-s + 0.953·13-s + 2.13·14-s − 0.123·15-s + 16-s − 1.40·17-s − 1.92·18-s − 19-s + 0.118·20-s − 2.21·21-s + 5.30·22-s − 7.01·23-s − 1.03·24-s − 4.98·25-s + 0.953·26-s + 5.10·27-s + 2.13·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.598·3-s + 0.5·4-s + 0.0531·5-s − 0.423·6-s + 0.806·7-s + 0.353·8-s − 0.642·9-s + 0.0375·10-s + 1.59·11-s − 0.299·12-s + 0.264·13-s + 0.570·14-s − 0.0317·15-s + 0.250·16-s − 0.341·17-s − 0.454·18-s − 0.229·19-s + 0.0265·20-s − 0.482·21-s + 1.13·22-s − 1.46·23-s − 0.211·24-s − 0.997·25-s + 0.186·26-s + 0.982·27-s + 0.403·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.082388616$
$L(\frac12)$  $\approx$  $3.082388616$
$L(\frac{3}{2})$   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 \( 1 + 1.03T + 3T^{2} \)
5 \( 1 - 0.118T + 5T^{2} \)
7 \( 1 - 2.13T + 7T^{2} \)
11 \( 1 - 5.30T + 11T^{2} \)
13 \( 1 - 0.953T + 13T^{2} \)
17 \( 1 + 1.40T + 17T^{2} \)
23 \( 1 + 7.01T + 23T^{2} \)
29 \( 1 + 3.22T + 29T^{2} \)
31 \( 1 - 5.86T + 31T^{2} \)
37 \( 1 - 9.27T + 37T^{2} \)
41 \( 1 - 6.08T + 41T^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 - 5.03T + 47T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 + 6.60T + 59T^{2} \)
61 \( 1 - 8.99T + 61T^{2} \)
67 \( 1 - 5.49T + 67T^{2} \)
71 \( 1 - 13.7T + 71T^{2} \)
73 \( 1 - 13.0T + 73T^{2} \)
79 \( 1 - 9.40T + 79T^{2} \)
83 \( 1 + 1.93T + 83T^{2} \)
89 \( 1 + 16.4T + 89T^{2} \)
97 \( 1 + 5.85T + 97T^{2} \)
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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

−7.968248464918044616332602389221, −6.86673564897997314336112450965, −6.10420836838673270004907306628, −5.98698595571795966952450930653, −5.03402137073591851813822816152, −4.15794757612550156636474685825, −3.92847014787813130684720873475, −2.64039603555761322898197604050, −1.84233272033713333202056703182, −0.826280832445870339785328152178, 0.826280832445870339785328152178, 1.84233272033713333202056703182, 2.64039603555761322898197604050, 3.92847014787813130684720873475, 4.15794757612550156636474685825, 5.03402137073591851813822816152, 5.98698595571795966952450930653, 6.10420836838673270004907306628, 6.86673564897997314336112450965, 7.968248464918044616332602389221

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