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 + 1.45·3-s + 4-s + 3.97·5-s + 1.45·6-s − 1.21·7-s + 8-s − 0.889·9-s + 3.97·10-s + 5.76·11-s + 1.45·12-s − 3.05·13-s − 1.21·14-s + 5.76·15-s + 16-s − 4.87·17-s − 0.889·18-s − 19-s + 3.97·20-s − 1.76·21-s + 5.76·22-s + 4.80·23-s + 1.45·24-s + 10.7·25-s − 3.05·26-s − 5.65·27-s − 1.21·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.838·3-s + 0.5·4-s + 1.77·5-s + 0.593·6-s − 0.458·7-s + 0.353·8-s − 0.296·9-s + 1.25·10-s + 1.73·11-s + 0.419·12-s − 0.846·13-s − 0.324·14-s + 1.48·15-s + 0.250·16-s − 1.18·17-s − 0.209·18-s − 0.229·19-s + 0.887·20-s − 0.384·21-s + 1.22·22-s + 1.00·23-s + 0.296·24-s + 2.15·25-s − 0.598·26-s − 1.08·27-s − 0.229·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.488763755$
$L(\frac12)$  $\approx$  $6.488763755$
$L(\frac{3}{2})$   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 \( 1 - 1.45T + 3T^{2} \)
5 \( 1 - 3.97T + 5T^{2} \)
7 \( 1 + 1.21T + 7T^{2} \)
11 \( 1 - 5.76T + 11T^{2} \)
13 \( 1 + 3.05T + 13T^{2} \)
17 \( 1 + 4.87T + 17T^{2} \)
23 \( 1 - 4.80T + 23T^{2} \)
29 \( 1 - 5.96T + 29T^{2} \)
31 \( 1 - 10.6T + 31T^{2} \)
37 \( 1 - 2.67T + 37T^{2} \)
41 \( 1 + 7.24T + 41T^{2} \)
43 \( 1 - 9.61T + 43T^{2} \)
47 \( 1 - 6.54T + 47T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 - 0.248T + 59T^{2} \)
61 \( 1 + 14.0T + 61T^{2} \)
67 \( 1 + 0.642T + 67T^{2} \)
71 \( 1 + 5.96T + 71T^{2} \)
73 \( 1 + 6.74T + 73T^{2} \)
79 \( 1 - 16.4T + 79T^{2} \)
83 \( 1 - 12.2T + 83T^{2} \)
89 \( 1 - 7.32T + 89T^{2} \)
97 \( 1 + 5.02T + 97T^{2} \)
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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

−7.78353643797567312261185895732, −6.68145887041892816288129481064, −6.48447000448233765872104243484, −5.95259130145173176868377402562, −4.87653821515927897459448879867, −4.43106074407745021468092257126, −3.26979113182773924577648316182, −2.68571557536084635231525525033, −2.10035905856866506740079706506, −1.16533827511722347768666403445, 1.16533827511722347768666403445, 2.10035905856866506740079706506, 2.68571557536084635231525525033, 3.26979113182773924577648316182, 4.43106074407745021468092257126, 4.87653821515927897459448879867, 5.95259130145173176868377402562, 6.48447000448233765872104243484, 6.68145887041892816288129481064, 7.78353643797567312261185895732

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