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 − 0.860·3-s + 4-s + 2.52·5-s + 0.860·6-s − 0.0296·7-s − 8-s − 2.25·9-s − 2.52·10-s + 5.01·11-s − 0.860·12-s − 0.222·13-s + 0.0296·14-s − 2.16·15-s + 16-s + 3.18·17-s + 2.25·18-s − 19-s + 2.52·20-s + 0.0255·21-s − 5.01·22-s + 5.25·23-s + 0.860·24-s + 1.35·25-s + 0.222·26-s + 4.52·27-s − 0.0296·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.496·3-s + 0.5·4-s + 1.12·5-s + 0.351·6-s − 0.0112·7-s − 0.353·8-s − 0.753·9-s − 0.796·10-s + 1.51·11-s − 0.248·12-s − 0.0617·13-s + 0.00792·14-s − 0.559·15-s + 0.250·16-s + 0.772·17-s + 0.532·18-s − 0.229·19-s + 0.563·20-s + 0.00557·21-s − 1.06·22-s + 1.09·23-s + 0.175·24-s + 0.270·25-s + 0.0436·26-s + 0.871·27-s − 0.00560·28-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 \( 1 + 0.860T + 3T^{2} \)
5 \( 1 - 2.52T + 5T^{2} \)
7 \( 1 + 0.0296T + 7T^{2} \)
11 \( 1 - 5.01T + 11T^{2} \)
13 \( 1 + 0.222T + 13T^{2} \)
17 \( 1 - 3.18T + 17T^{2} \)
23 \( 1 - 5.25T + 23T^{2} \)
29 \( 1 + 9.76T + 29T^{2} \)
31 \( 1 + 1.46T + 31T^{2} \)
37 \( 1 + 8.01T + 37T^{2} \)
41 \( 1 + 11.7T + 41T^{2} \)
43 \( 1 - 1.24T + 43T^{2} \)
47 \( 1 + 6.14T + 47T^{2} \)
53 \( 1 + 1.12T + 53T^{2} \)
59 \( 1 + 8.52T + 59T^{2} \)
61 \( 1 - 3.33T + 61T^{2} \)
67 \( 1 - 0.833T + 67T^{2} \)
71 \( 1 - 5.44T + 71T^{2} \)
73 \( 1 + 0.905T + 73T^{2} \)
79 \( 1 + 6.25T + 79T^{2} \)
83 \( 1 + 10.6T + 83T^{2} \)
89 \( 1 + 12.0T + 89T^{2} \)
97 \( 1 + 9.41T + 97T^{2} \)
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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

−7.36473544666075613901212245006, −6.71708551372416913820007148852, −6.20009239386815536329264356667, −5.53486089284593635388310595494, −5.00033766820856805980675829487, −3.69882264533533757936450335154, −3.03876077699876304468173726217, −1.84051934343752353508966270165, −1.36702062118188758086692308137, 0, 1.36702062118188758086692308137, 1.84051934343752353508966270165, 3.03876077699876304468173726217, 3.69882264533533757936450335154, 5.00033766820856805980675829487, 5.53486089284593635388310595494, 6.20009239386815536329264356667, 6.71708551372416913820007148852, 7.36473544666075613901212245006

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