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 − 2.39·3-s + 4-s − 3.46·5-s − 2.39·6-s − 3.11·7-s + 8-s + 2.73·9-s − 3.46·10-s − 4.76·11-s − 2.39·12-s − 3.60·13-s − 3.11·14-s + 8.30·15-s + 16-s + 2.91·17-s + 2.73·18-s + 19-s − 3.46·20-s + 7.46·21-s − 4.76·22-s + 4.74·23-s − 2.39·24-s + 7.01·25-s − 3.60·26-s + 0.624·27-s − 3.11·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.38·3-s + 0.5·4-s − 1.55·5-s − 0.978·6-s − 1.17·7-s + 0.353·8-s + 0.913·9-s − 1.09·10-s − 1.43·11-s − 0.691·12-s − 0.998·13-s − 0.832·14-s + 2.14·15-s + 0.250·16-s + 0.707·17-s + 0.645·18-s + 0.229·19-s − 0.775·20-s + 1.62·21-s − 1.01·22-s + 0.989·23-s − 0.489·24-s + 1.40·25-s − 0.706·26-s + 0.120·27-s − 0.588·28-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 \( 1 + 2.39T + 3T^{2} \)
5 \( 1 + 3.46T + 5T^{2} \)
7 \( 1 + 3.11T + 7T^{2} \)
11 \( 1 + 4.76T + 11T^{2} \)
13 \( 1 + 3.60T + 13T^{2} \)
17 \( 1 - 2.91T + 17T^{2} \)
23 \( 1 - 4.74T + 23T^{2} \)
29 \( 1 + 4.17T + 29T^{2} \)
31 \( 1 - 6.99T + 31T^{2} \)
37 \( 1 - 0.716T + 37T^{2} \)
41 \( 1 + 3.08T + 41T^{2} \)
43 \( 1 - 0.576T + 43T^{2} \)
47 \( 1 - 1.44T + 47T^{2} \)
53 \( 1 - 13.7T + 53T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 + 1.24T + 61T^{2} \)
67 \( 1 + 0.735T + 67T^{2} \)
71 \( 1 + 15.1T + 71T^{2} \)
73 \( 1 - 5.48T + 73T^{2} \)
79 \( 1 + 3.39T + 79T^{2} \)
83 \( 1 + 9.99T + 83T^{2} \)
89 \( 1 - 6.54T + 89T^{2} \)
97 \( 1 - 3.66T + 97T^{2} \)
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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

−7.22885834302014696442543557832, −6.87092865816249762480284995016, −5.92414372918471504508804057226, −5.32654658221874643514780511694, −4.81209850671009207866682662981, −4.04748122770274796378839457771, −3.17593509897632323641492360907, −2.64395039297522240861736836196, −0.77475302560310758174189215675, 0, 0.77475302560310758174189215675, 2.64395039297522240861736836196, 3.17593509897632323641492360907, 4.04748122770274796378839457771, 4.81209850671009207866682662981, 5.32654658221874643514780511694, 5.92414372918471504508804057226, 6.87092865816249762480284995016, 7.22885834302014696442543557832

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