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 − 1.11·3-s + 4-s − 2.63·5-s + 1.11·6-s − 4.04·7-s − 8-s − 1.76·9-s + 2.63·10-s − 0.240·11-s − 1.11·12-s − 4.90·13-s + 4.04·14-s + 2.93·15-s + 16-s − 2.68·17-s + 1.76·18-s − 19-s − 2.63·20-s + 4.49·21-s + 0.240·22-s + 4.56·23-s + 1.11·24-s + 1.95·25-s + 4.90·26-s + 5.29·27-s − 4.04·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.642·3-s + 0.5·4-s − 1.17·5-s + 0.454·6-s − 1.52·7-s − 0.353·8-s − 0.587·9-s + 0.834·10-s − 0.0723·11-s − 0.321·12-s − 1.35·13-s + 1.08·14-s + 0.757·15-s + 0.250·16-s − 0.650·17-s + 0.415·18-s − 0.229·19-s − 0.589·20-s + 0.981·21-s + 0.0511·22-s + 0.950·23-s + 0.227·24-s + 0.391·25-s + 0.961·26-s + 1.01·27-s − 0.763·28-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 \( 1 + 1.11T + 3T^{2} \)
5 \( 1 + 2.63T + 5T^{2} \)
7 \( 1 + 4.04T + 7T^{2} \)
11 \( 1 + 0.240T + 11T^{2} \)
13 \( 1 + 4.90T + 13T^{2} \)
17 \( 1 + 2.68T + 17T^{2} \)
23 \( 1 - 4.56T + 23T^{2} \)
29 \( 1 + 3.19T + 29T^{2} \)
31 \( 1 - 4.37T + 31T^{2} \)
37 \( 1 + 9.88T + 37T^{2} \)
41 \( 1 + 0.240T + 41T^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 - 5.99T + 47T^{2} \)
53 \( 1 + 11.9T + 53T^{2} \)
59 \( 1 + 3.67T + 59T^{2} \)
61 \( 1 - 4.10T + 61T^{2} \)
67 \( 1 - 13.1T + 67T^{2} \)
71 \( 1 - 6.19T + 71T^{2} \)
73 \( 1 - 0.320T + 73T^{2} \)
79 \( 1 - 3.48T + 79T^{2} \)
83 \( 1 - 1.60T + 83T^{2} \)
89 \( 1 - 1.30T + 89T^{2} \)
97 \( 1 + 15.2T + 97T^{2} \)
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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

−7.40444357107755728199389842839, −6.87067106761113287815653888705, −6.35637040665767656216769496043, −5.48862194786259432532128687461, −4.72385342721215917125783495522, −3.77173064684126166156672573543, −3.03000154402076250805665005890, −2.34881105300437848894740270927, −0.62324847783582474252169983077, 0, 0.62324847783582474252169983077, 2.34881105300437848894740270927, 3.03000154402076250805665005890, 3.77173064684126166156672573543, 4.72385342721215917125783495522, 5.48862194786259432532128687461, 6.35637040665767656216769496043, 6.87067106761113287815653888705, 7.40444357107755728199389842839

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