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 − 0.656·3-s + 4-s − 1.22·5-s + 0.656·6-s + 2.46·7-s − 8-s − 2.56·9-s + 1.22·10-s + 2.53·11-s − 0.656·12-s + 2.77·13-s − 2.46·14-s + 0.806·15-s + 16-s − 0.964·17-s + 2.56·18-s − 19-s − 1.22·20-s − 1.61·21-s − 2.53·22-s + 3.35·23-s + 0.656·24-s − 3.48·25-s − 2.77·26-s + 3.65·27-s + 2.46·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.378·3-s + 0.5·4-s − 0.549·5-s + 0.267·6-s + 0.932·7-s − 0.353·8-s − 0.856·9-s + 0.388·10-s + 0.763·11-s − 0.189·12-s + 0.770·13-s − 0.659·14-s + 0.208·15-s + 0.250·16-s − 0.233·17-s + 0.605·18-s − 0.229·19-s − 0.274·20-s − 0.353·21-s − 0.540·22-s + 0.699·23-s + 0.133·24-s − 0.697·25-s − 0.544·26-s + 0.703·27-s + 0.466·28-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 \( 1 + 0.656T + 3T^{2} \)
5 \( 1 + 1.22T + 5T^{2} \)
7 \( 1 - 2.46T + 7T^{2} \)
11 \( 1 - 2.53T + 11T^{2} \)
13 \( 1 - 2.77T + 13T^{2} \)
17 \( 1 + 0.964T + 17T^{2} \)
23 \( 1 - 3.35T + 23T^{2} \)
29 \( 1 - 0.507T + 29T^{2} \)
31 \( 1 - 5.13T + 31T^{2} \)
37 \( 1 + 8.07T + 37T^{2} \)
41 \( 1 + 1.46T + 41T^{2} \)
43 \( 1 + 12.6T + 43T^{2} \)
47 \( 1 + 2.57T + 47T^{2} \)
53 \( 1 + 1.67T + 53T^{2} \)
59 \( 1 + 0.559T + 59T^{2} \)
61 \( 1 + 3.22T + 61T^{2} \)
67 \( 1 - 4.20T + 67T^{2} \)
71 \( 1 - 4.84T + 71T^{2} \)
73 \( 1 + 8.59T + 73T^{2} \)
79 \( 1 - 15.0T + 79T^{2} \)
83 \( 1 - 14.5T + 83T^{2} \)
89 \( 1 + 15.3T + 89T^{2} \)
97 \( 1 - 2.84T + 97T^{2} \)
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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

−7.69741135282082785896651625650, −6.66979337011114762316171676338, −6.40142053225896605630806115365, −5.37992250521533231100987384975, −4.79340196044929899407438161801, −3.81382128238763811782007634786, −3.11349235845796247636539052395, −1.95366095900967865780194539798, −1.15496957855666958717773835475, 0, 1.15496957855666958717773835475, 1.95366095900967865780194539798, 3.11349235845796247636539052395, 3.81382128238763811782007634786, 4.79340196044929899407438161801, 5.37992250521533231100987384975, 6.40142053225896605630806115365, 6.66979337011114762316171676338, 7.69741135282082785896651625650

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