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 − 3.23·3-s + 4-s − 3.65·5-s − 3.23·6-s + 2.39·7-s + 8-s + 7.43·9-s − 3.65·10-s + 0.229·11-s − 3.23·12-s − 1.51·13-s + 2.39·14-s + 11.7·15-s + 16-s + 1.23·17-s + 7.43·18-s + 19-s − 3.65·20-s − 7.72·21-s + 0.229·22-s − 8.59·23-s − 3.23·24-s + 8.33·25-s − 1.51·26-s − 14.3·27-s + 2.39·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.86·3-s + 0.5·4-s − 1.63·5-s − 1.31·6-s + 0.903·7-s + 0.353·8-s + 2.47·9-s − 1.15·10-s + 0.0690·11-s − 0.932·12-s − 0.421·13-s + 0.638·14-s + 3.04·15-s + 0.250·16-s + 0.299·17-s + 1.75·18-s + 0.229·19-s − 0.816·20-s − 1.68·21-s + 0.0488·22-s − 1.79·23-s − 0.659·24-s + 1.66·25-s − 0.298·26-s − 2.75·27-s + 0.451·28-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 \( 1 + 3.23T + 3T^{2} \)
5 \( 1 + 3.65T + 5T^{2} \)
7 \( 1 - 2.39T + 7T^{2} \)
11 \( 1 - 0.229T + 11T^{2} \)
13 \( 1 + 1.51T + 13T^{2} \)
17 \( 1 - 1.23T + 17T^{2} \)
23 \( 1 + 8.59T + 23T^{2} \)
29 \( 1 + 2.80T + 29T^{2} \)
31 \( 1 - 4.09T + 31T^{2} \)
37 \( 1 + 7.69T + 37T^{2} \)
41 \( 1 - 6.97T + 41T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 - 2.41T + 53T^{2} \)
59 \( 1 - 5.91T + 59T^{2} \)
61 \( 1 - 5.03T + 61T^{2} \)
67 \( 1 + 1.27T + 67T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 + 5.30T + 73T^{2} \)
79 \( 1 - 4.63T + 79T^{2} \)
83 \( 1 - 0.441T + 83T^{2} \)
89 \( 1 - 3.69T + 89T^{2} \)
97 \( 1 - 7.94T + 97T^{2} \)
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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

−7.41724150179195757658549749926, −6.70838925485580000849032672283, −5.97410747567096320731572833166, −5.22496516161833556153775025827, −4.78352244487710473195868917268, −4.08282823355455037042559429058, −3.63671893774816011824181872326, −2.10324036713783251922239193351, −1.00179035525224963913414570963, 0, 1.00179035525224963913414570963, 2.10324036713783251922239193351, 3.63671893774816011824181872326, 4.08282823355455037042559429058, 4.78352244487710473195868917268, 5.22496516161833556153775025827, 5.97410747567096320731572833166, 6.70838925485580000849032672283, 7.41724150179195757658549749926

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