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 + 2.23·3-s + 4-s + 0.346·5-s − 2.23·6-s + 1.27·7-s − 8-s + 2.00·9-s − 0.346·10-s − 4.01·11-s + 2.23·12-s − 1.58·13-s − 1.27·14-s + 0.774·15-s + 16-s + 4.76·17-s − 2.00·18-s − 19-s + 0.346·20-s + 2.84·21-s + 4.01·22-s − 3.55·23-s − 2.23·24-s − 4.88·25-s + 1.58·26-s − 2.22·27-s + 1.27·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.29·3-s + 0.5·4-s + 0.154·5-s − 0.913·6-s + 0.480·7-s − 0.353·8-s + 0.667·9-s − 0.109·10-s − 1.20·11-s + 0.645·12-s − 0.439·13-s − 0.339·14-s + 0.200·15-s + 0.250·16-s + 1.15·17-s − 0.472·18-s − 0.229·19-s + 0.0774·20-s + 0.620·21-s + 0.855·22-s − 0.741·23-s − 0.456·24-s − 0.976·25-s + 0.310·26-s − 0.428·27-s + 0.240·28-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 \( 1 - 2.23T + 3T^{2} \)
5 \( 1 - 0.346T + 5T^{2} \)
7 \( 1 - 1.27T + 7T^{2} \)
11 \( 1 + 4.01T + 11T^{2} \)
13 \( 1 + 1.58T + 13T^{2} \)
17 \( 1 - 4.76T + 17T^{2} \)
23 \( 1 + 3.55T + 23T^{2} \)
29 \( 1 + 5.97T + 29T^{2} \)
31 \( 1 - 5.18T + 31T^{2} \)
37 \( 1 - 7.85T + 37T^{2} \)
41 \( 1 - 3.23T + 41T^{2} \)
43 \( 1 + 12.4T + 43T^{2} \)
47 \( 1 + 2.70T + 47T^{2} \)
53 \( 1 - 12.5T + 53T^{2} \)
59 \( 1 - 1.46T + 59T^{2} \)
61 \( 1 - 0.631T + 61T^{2} \)
67 \( 1 + 4.91T + 67T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 + 9.52T + 73T^{2} \)
79 \( 1 + 2.20T + 79T^{2} \)
83 \( 1 + 2.94T + 83T^{2} \)
89 \( 1 + 18.2T + 89T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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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

−7.60841779370843533591913164045, −7.36857353986171329177668163983, −6.08146771200636474159757320685, −5.53276090040751747760928647963, −4.57953702762925238957377390535, −3.64213293334153813081837855177, −2.85511450774194715035402351435, −2.26156685914355343397513763065, −1.48964382956637716524617843279, 0, 1.48964382956637716524617843279, 2.26156685914355343397513763065, 2.85511450774194715035402351435, 3.64213293334153813081837855177, 4.57953702762925238957377390535, 5.53276090040751747760928647963, 6.08146771200636474159757320685, 7.36857353986171329177668163983, 7.60841779370843533591913164045

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