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 − 2.04·3-s + 4-s − 4.09·5-s + 2.04·6-s + 1.49·7-s − 8-s + 1.18·9-s + 4.09·10-s + 4.36·11-s − 2.04·12-s − 2.07·13-s − 1.49·14-s + 8.37·15-s + 16-s + 2.71·17-s − 1.18·18-s − 19-s − 4.09·20-s − 3.06·21-s − 4.36·22-s − 8.72·23-s + 2.04·24-s + 11.7·25-s + 2.07·26-s + 3.71·27-s + 1.49·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.18·3-s + 0.5·4-s − 1.83·5-s + 0.835·6-s + 0.566·7-s − 0.353·8-s + 0.395·9-s + 1.29·10-s + 1.31·11-s − 0.590·12-s − 0.576·13-s − 0.400·14-s + 2.16·15-s + 0.250·16-s + 0.658·17-s − 0.279·18-s − 0.229·19-s − 0.915·20-s − 0.668·21-s − 0.930·22-s − 1.81·23-s + 0.417·24-s + 2.35·25-s + 0.407·26-s + 0.714·27-s + 0.283·28-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 \( 1 + 2.04T + 3T^{2} \)
5 \( 1 + 4.09T + 5T^{2} \)
7 \( 1 - 1.49T + 7T^{2} \)
11 \( 1 - 4.36T + 11T^{2} \)
13 \( 1 + 2.07T + 13T^{2} \)
17 \( 1 - 2.71T + 17T^{2} \)
23 \( 1 + 8.72T + 23T^{2} \)
29 \( 1 - 1.76T + 29T^{2} \)
31 \( 1 + 3.93T + 31T^{2} \)
37 \( 1 + 9.86T + 37T^{2} \)
41 \( 1 - 5.20T + 41T^{2} \)
43 \( 1 - 4.35T + 43T^{2} \)
47 \( 1 + 5.27T + 47T^{2} \)
53 \( 1 + 2.90T + 53T^{2} \)
59 \( 1 - 7.50T + 59T^{2} \)
61 \( 1 + 4.68T + 61T^{2} \)
67 \( 1 + 8.57T + 67T^{2} \)
71 \( 1 - 1.09T + 71T^{2} \)
73 \( 1 - 11.3T + 73T^{2} \)
79 \( 1 - 7.46T + 79T^{2} \)
83 \( 1 - 15.3T + 83T^{2} \)
89 \( 1 - 7.24T + 89T^{2} \)
97 \( 1 + 16.3T + 97T^{2} \)
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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

−7.63067013637624980659618813646, −6.83035511134184003671617893336, −6.32781066246640520788511568340, −5.41127269962562439250556124571, −4.64467724394778478550893568773, −3.96728104202722070213390645597, −3.29598020292521397541170974860, −1.87963281992997566057202134114, −0.828515127013870510752212403165, 0, 0.828515127013870510752212403165, 1.87963281992997566057202134114, 3.29598020292521397541170974860, 3.96728104202722070213390645597, 4.64467724394778478550893568773, 5.41127269962562439250556124571, 6.32781066246640520788511568340, 6.83035511134184003671617893336, 7.63067013637624980659618813646

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