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 + 2.77·3-s + 4-s + 0.377·5-s − 2.77·6-s + 0.478·7-s − 8-s + 4.67·9-s − 0.377·10-s − 4.92·11-s + 2.77·12-s − 1.38·13-s − 0.478·14-s + 1.04·15-s + 16-s − 5.72·17-s − 4.67·18-s − 19-s + 0.377·20-s + 1.32·21-s + 4.92·22-s + 7.43·23-s − 2.77·24-s − 4.85·25-s + 1.38·26-s + 4.63·27-s + 0.478·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.59·3-s + 0.5·4-s + 0.169·5-s − 1.13·6-s + 0.180·7-s − 0.353·8-s + 1.55·9-s − 0.119·10-s − 1.48·11-s + 0.799·12-s − 0.383·13-s − 0.127·14-s + 0.270·15-s + 0.250·16-s − 1.38·17-s − 1.10·18-s − 0.229·19-s + 0.0845·20-s + 0.289·21-s + 1.05·22-s + 1.55·23-s − 0.565·24-s − 0.971·25-s + 0.271·26-s + 0.892·27-s + 0.0903·28-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 \( 1 - 2.77T + 3T^{2} \)
5 \( 1 - 0.377T + 5T^{2} \)
7 \( 1 - 0.478T + 7T^{2} \)
11 \( 1 + 4.92T + 11T^{2} \)
13 \( 1 + 1.38T + 13T^{2} \)
17 \( 1 + 5.72T + 17T^{2} \)
23 \( 1 - 7.43T + 23T^{2} \)
29 \( 1 - 5.15T + 29T^{2} \)
31 \( 1 + 0.491T + 31T^{2} \)
37 \( 1 - 0.0388T + 37T^{2} \)
41 \( 1 + 0.959T + 41T^{2} \)
43 \( 1 + 6.69T + 43T^{2} \)
47 \( 1 - 1.82T + 47T^{2} \)
53 \( 1 + 6.62T + 53T^{2} \)
59 \( 1 + 1.37T + 59T^{2} \)
61 \( 1 - 10.0T + 61T^{2} \)
67 \( 1 + 4.85T + 67T^{2} \)
71 \( 1 + 5.45T + 71T^{2} \)
73 \( 1 - 4.45T + 73T^{2} \)
79 \( 1 - 0.976T + 79T^{2} \)
83 \( 1 - 15.4T + 83T^{2} \)
89 \( 1 + 0.324T + 89T^{2} \)
97 \( 1 + 14.0T + 97T^{2} \)
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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

−7.85774120811771810983681177659, −7.03212229572036243252842727818, −6.46886125320221581238284038629, −5.25055976235591222684814820899, −4.64734196400537702617720817259, −3.61643462418144777952568976144, −2.71672762212899329116987233617, −2.42098965008141248572360697896, −1.52054441727227531419663054180, 0, 1.52054441727227531419663054180, 2.42098965008141248572360697896, 2.71672762212899329116987233617, 3.61643462418144777952568976144, 4.64734196400537702617720817259, 5.25055976235591222684814820899, 6.46886125320221581238284038629, 7.03212229572036243252842727818, 7.85774120811771810983681177659

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