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 + 1.50·3-s + 4-s + 0.477·5-s − 1.50·6-s + 1.81·7-s − 8-s − 0.742·9-s − 0.477·10-s + 3.72·11-s + 1.50·12-s − 2.78·13-s − 1.81·14-s + 0.717·15-s + 16-s − 6.65·17-s + 0.742·18-s − 19-s + 0.477·20-s + 2.73·21-s − 3.72·22-s − 9.04·23-s − 1.50·24-s − 4.77·25-s + 2.78·26-s − 5.62·27-s + 1.81·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.867·3-s + 0.5·4-s + 0.213·5-s − 0.613·6-s + 0.687·7-s − 0.353·8-s − 0.247·9-s − 0.150·10-s + 1.12·11-s + 0.433·12-s − 0.773·13-s − 0.486·14-s + 0.185·15-s + 0.250·16-s − 1.61·17-s + 0.175·18-s − 0.229·19-s + 0.106·20-s + 0.596·21-s − 0.795·22-s − 1.88·23-s − 0.306·24-s − 0.954·25-s + 0.546·26-s − 1.08·27-s + 0.343·28-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 \( 1 - 1.50T + 3T^{2} \)
5 \( 1 - 0.477T + 5T^{2} \)
7 \( 1 - 1.81T + 7T^{2} \)
11 \( 1 - 3.72T + 11T^{2} \)
13 \( 1 + 2.78T + 13T^{2} \)
17 \( 1 + 6.65T + 17T^{2} \)
23 \( 1 + 9.04T + 23T^{2} \)
29 \( 1 - 10.2T + 29T^{2} \)
31 \( 1 - 7.70T + 31T^{2} \)
37 \( 1 + 2.81T + 37T^{2} \)
41 \( 1 - 0.494T + 41T^{2} \)
43 \( 1 + 3.56T + 43T^{2} \)
47 \( 1 - 7.49T + 47T^{2} \)
53 \( 1 - 10.9T + 53T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 - 3.68T + 61T^{2} \)
67 \( 1 - 2.58T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 + 1.74T + 73T^{2} \)
79 \( 1 + 0.328T + 79T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 + 13.0T + 89T^{2} \)
97 \( 1 + 8.73T + 97T^{2} \)
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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

−7.77675210564404689841650784520, −6.80067479266639871848993418886, −6.40013478959333578360751850700, −5.50731407621849232432692987778, −4.39464510800560208989435547180, −3.98947794573706052565469704499, −2.70045298086073995894973573423, −2.25662690906465155669447215952, −1.43643358916812911765849969472, 0, 1.43643358916812911765849969472, 2.25662690906465155669447215952, 2.70045298086073995894973573423, 3.98947794573706052565469704499, 4.39464510800560208989435547180, 5.50731407621849232432692987778, 6.40013478959333578360751850700, 6.80067479266639871848993418886, 7.77675210564404689841650784520

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