Properties

Degree 2
Conductor $ 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  − 2.66·2-s − 1.96·3-s + 5.11·4-s − 2.76·5-s + 5.23·6-s + 2.50·7-s − 8.29·8-s + 0.857·9-s + 7.37·10-s − 4.45·11-s − 10.0·12-s − 2.71·13-s − 6.67·14-s + 5.43·15-s + 11.8·16-s − 6.37·17-s − 2.28·18-s + 19-s − 14.1·20-s − 4.91·21-s + 11.8·22-s + 4.98·23-s + 16.2·24-s + 2.64·25-s + 7.22·26-s + 4.20·27-s + 12.7·28-s + ⋯
L(s)  = 1  − 1.88·2-s − 1.13·3-s + 2.55·4-s − 1.23·5-s + 2.13·6-s + 0.945·7-s − 2.93·8-s + 0.285·9-s + 2.33·10-s − 1.34·11-s − 2.89·12-s − 0.751·13-s − 1.78·14-s + 1.40·15-s + 2.97·16-s − 1.54·17-s − 0.538·18-s + 0.229·19-s − 3.15·20-s − 1.07·21-s + 2.53·22-s + 1.04·23-s + 3.32·24-s + 0.528·25-s + 1.41·26-s + 0.809·27-s + 2.41·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4009\)    =    \(19 \cdot 211\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4009} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4009,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{19,\;211\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{19,\;211\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad19 \( 1 - T \)
211 \( 1 - T \)
good2 \( 1 + 2.66T + 2T^{2} \)
3 \( 1 + 1.96T + 3T^{2} \)
5 \( 1 + 2.76T + 5T^{2} \)
7 \( 1 - 2.50T + 7T^{2} \)
11 \( 1 + 4.45T + 11T^{2} \)
13 \( 1 + 2.71T + 13T^{2} \)
17 \( 1 + 6.37T + 17T^{2} \)
23 \( 1 - 4.98T + 23T^{2} \)
29 \( 1 - 6.14T + 29T^{2} \)
31 \( 1 + 4.08T + 31T^{2} \)
37 \( 1 + 0.469T + 37T^{2} \)
41 \( 1 + 6.42T + 41T^{2} \)
43 \( 1 - 9.76T + 43T^{2} \)
47 \( 1 - 13.1T + 47T^{2} \)
53 \( 1 + 12.3T + 53T^{2} \)
59 \( 1 - 8.81T + 59T^{2} \)
61 \( 1 + 10.6T + 61T^{2} \)
67 \( 1 - 5.32T + 67T^{2} \)
71 \( 1 + 14.9T + 71T^{2} \)
73 \( 1 - 2.77T + 73T^{2} \)
79 \( 1 - 4.85T + 79T^{2} \)
83 \( 1 - 14.7T + 83T^{2} \)
89 \( 1 - 6.53T + 89T^{2} \)
97 \( 1 - 8.97T + 97T^{2} \)
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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

−8.113358754820247166141541066844, −7.45853748770901399513957602062, −7.05485678935525354451380306036, −6.14388080125945713967668865446, −5.13205433352554440066383234118, −4.54964365678768302614310579392, −2.97763215824120928439466361285, −2.12936691288108114081086015983, −0.76772656808138340078504400697, 0, 0.76772656808138340078504400697, 2.12936691288108114081086015983, 2.97763215824120928439466361285, 4.54964365678768302614310579392, 5.13205433352554440066383234118, 6.14388080125945713967668865446, 7.05485678935525354451380306036, 7.45853748770901399513957602062, 8.113358754820247166141541066844

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