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 + 1.86·3-s + 4-s − 2.09·5-s − 1.86·6-s + 2.42·7-s − 8-s + 0.487·9-s + 2.09·10-s + 2.99·11-s + 1.86·12-s − 5.37·13-s − 2.42·14-s − 3.91·15-s + 16-s + 0.964·17-s − 0.487·18-s − 19-s − 2.09·20-s + 4.53·21-s − 2.99·22-s + 3.99·23-s − 1.86·24-s − 0.615·25-s + 5.37·26-s − 4.69·27-s + 2.42·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.07·3-s + 0.5·4-s − 0.936·5-s − 0.762·6-s + 0.917·7-s − 0.353·8-s + 0.162·9-s + 0.662·10-s + 0.902·11-s + 0.539·12-s − 1.49·13-s − 0.648·14-s − 1.00·15-s + 0.250·16-s + 0.233·17-s − 0.114·18-s − 0.229·19-s − 0.468·20-s + 0.989·21-s − 0.638·22-s + 0.832·23-s − 0.381·24-s − 0.123·25-s + 1.05·26-s − 0.903·27-s + 0.458·28-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 \( 1 - 1.86T + 3T^{2} \)
5 \( 1 + 2.09T + 5T^{2} \)
7 \( 1 - 2.42T + 7T^{2} \)
11 \( 1 - 2.99T + 11T^{2} \)
13 \( 1 + 5.37T + 13T^{2} \)
17 \( 1 - 0.964T + 17T^{2} \)
23 \( 1 - 3.99T + 23T^{2} \)
29 \( 1 + 0.375T + 29T^{2} \)
31 \( 1 + 4.16T + 31T^{2} \)
37 \( 1 - 7.39T + 37T^{2} \)
41 \( 1 - 0.401T + 41T^{2} \)
43 \( 1 - 1.78T + 43T^{2} \)
47 \( 1 + 4.63T + 47T^{2} \)
53 \( 1 + 8.02T + 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 - 12.8T + 61T^{2} \)
67 \( 1 - 10.4T + 67T^{2} \)
71 \( 1 + 8.96T + 71T^{2} \)
73 \( 1 - 3.29T + 73T^{2} \)
79 \( 1 + 15.5T + 79T^{2} \)
83 \( 1 - 0.126T + 83T^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 + 16.1T + 97T^{2} \)
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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

−7.56817710189475569719541305807, −7.30992090000583085282508675361, −6.35865162616602020956127879605, −5.29124927572909872104820483853, −4.50584991318164965225938417021, −3.78398256576307826038446149799, −2.96865620683968879357230114326, −2.22090416200052067532163426854, −1.33877795295000928203890386999, 0, 1.33877795295000928203890386999, 2.22090416200052067532163426854, 2.96865620683968879357230114326, 3.78398256576307826038446149799, 4.50584991318164965225938417021, 5.29124927572909872104820483853, 6.35865162616602020956127879605, 7.30992090000583085282508675361, 7.56817710189475569719541305807

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