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 + 3.26·3-s + 4-s + 0.595·5-s − 3.26·6-s − 2.99·7-s − 8-s + 7.67·9-s − 0.595·10-s + 0.495·11-s + 3.26·12-s − 2.84·13-s + 2.99·14-s + 1.94·15-s + 16-s − 1.90·17-s − 7.67·18-s − 19-s + 0.595·20-s − 9.77·21-s − 0.495·22-s − 8.88·23-s − 3.26·24-s − 4.64·25-s + 2.84·26-s + 15.2·27-s − 2.99·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.88·3-s + 0.5·4-s + 0.266·5-s − 1.33·6-s − 1.13·7-s − 0.353·8-s + 2.55·9-s − 0.188·10-s + 0.149·11-s + 0.943·12-s − 0.789·13-s + 0.799·14-s + 0.502·15-s + 0.250·16-s − 0.461·17-s − 1.80·18-s − 0.229·19-s + 0.133·20-s − 2.13·21-s − 0.105·22-s − 1.85·23-s − 0.667·24-s − 0.929·25-s + 0.558·26-s + 2.94·27-s − 0.565·28-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 \( 1 - 3.26T + 3T^{2} \)
5 \( 1 - 0.595T + 5T^{2} \)
7 \( 1 + 2.99T + 7T^{2} \)
11 \( 1 - 0.495T + 11T^{2} \)
13 \( 1 + 2.84T + 13T^{2} \)
17 \( 1 + 1.90T + 17T^{2} \)
23 \( 1 + 8.88T + 23T^{2} \)
29 \( 1 + 0.0817T + 29T^{2} \)
31 \( 1 + 4.72T + 31T^{2} \)
37 \( 1 + 1.10T + 37T^{2} \)
41 \( 1 - 8.82T + 41T^{2} \)
43 \( 1 - 4.95T + 43T^{2} \)
47 \( 1 + 2.10T + 47T^{2} \)
53 \( 1 - 0.933T + 53T^{2} \)
59 \( 1 - 10.6T + 59T^{2} \)
61 \( 1 - 6.78T + 61T^{2} \)
67 \( 1 + 3.28T + 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
79 \( 1 + 3.99T + 79T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 + 11.0T + 89T^{2} \)
97 \( 1 + 6.88T + 97T^{2} \)
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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

−7.66296283021669515830550627832, −7.07260659425530615471404272732, −6.41747212101862587240939116196, −5.57246092077319608905017352056, −4.10870645929102387504647022089, −3.86981426497806290566136469578, −2.76031635301284324073435244115, −2.37961309095389551026327935975, −1.57419353894283704324174263052, 0, 1.57419353894283704324174263052, 2.37961309095389551026327935975, 2.76031635301284324073435244115, 3.86981426497806290566136469578, 4.10870645929102387504647022089, 5.57246092077319608905017352056, 6.41747212101862587240939116196, 7.07260659425530615471404272732, 7.66296283021669515830550627832

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