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 − 1.46·3-s + 4-s + 2.14·5-s − 1.46·6-s + 2.84·7-s + 8-s − 0.862·9-s + 2.14·10-s − 4.14·11-s − 1.46·12-s + 5.72·13-s + 2.84·14-s − 3.13·15-s + 16-s + 0.828·17-s − 0.862·18-s − 19-s + 2.14·20-s − 4.16·21-s − 4.14·22-s + 4.11·23-s − 1.46·24-s − 0.402·25-s + 5.72·26-s + 5.64·27-s + 2.84·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.844·3-s + 0.5·4-s + 0.958·5-s − 0.596·6-s + 1.07·7-s + 0.353·8-s − 0.287·9-s + 0.678·10-s − 1.24·11-s − 0.422·12-s + 1.58·13-s + 0.760·14-s − 0.809·15-s + 0.250·16-s + 0.200·17-s − 0.203·18-s − 0.229·19-s + 0.479·20-s − 0.907·21-s − 0.883·22-s + 0.857·23-s − 0.298·24-s − 0.0804·25-s + 1.12·26-s + 1.08·27-s + 0.537·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.430180788$
$L(\frac12)$  $\approx$  $3.430180788$
$L(\frac{3}{2})$   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 \( 1 + 1.46T + 3T^{2} \)
5 \( 1 - 2.14T + 5T^{2} \)
7 \( 1 - 2.84T + 7T^{2} \)
11 \( 1 + 4.14T + 11T^{2} \)
13 \( 1 - 5.72T + 13T^{2} \)
17 \( 1 - 0.828T + 17T^{2} \)
23 \( 1 - 4.11T + 23T^{2} \)
29 \( 1 - 5.07T + 29T^{2} \)
31 \( 1 - 3.20T + 31T^{2} \)
37 \( 1 + 1.30T + 37T^{2} \)
41 \( 1 + 9.49T + 41T^{2} \)
43 \( 1 - 1.76T + 43T^{2} \)
47 \( 1 - 6.02T + 47T^{2} \)
53 \( 1 + 9.71T + 53T^{2} \)
59 \( 1 + 1.16T + 59T^{2} \)
61 \( 1 + 0.741T + 61T^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 - 1.20T + 71T^{2} \)
73 \( 1 - 14.2T + 73T^{2} \)
79 \( 1 - 0.0154T + 79T^{2} \)
83 \( 1 - 5.25T + 83T^{2} \)
89 \( 1 - 15.8T + 89T^{2} \)
97 \( 1 + 9.00T + 97T^{2} \)
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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

−7.967361426339482163444432663674, −6.74217715791830328946530845776, −6.30075417046773193237007317989, −5.58768095731770946084402119721, −5.16237534190177365854296313081, −4.66184817816416869669320045040, −3.52405043420001991979971600856, −2.66809401701116827664512414800, −1.80458160152447498757900666323, −0.895640247847869999687669837710, 0.895640247847869999687669837710, 1.80458160152447498757900666323, 2.66809401701116827664512414800, 3.52405043420001991979971600856, 4.66184817816416869669320045040, 5.16237534190177365854296313081, 5.58768095731770946084402119721, 6.30075417046773193237007317989, 6.74217715791830328946530845776, 7.967361426339482163444432663674

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