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 + 2.75·3-s + 4-s + 1.27·5-s − 2.75·6-s + 0.521·7-s − 8-s + 4.61·9-s − 1.27·10-s − 2.91·11-s + 2.75·12-s − 1.67·13-s − 0.521·14-s + 3.52·15-s + 16-s − 1.41·17-s − 4.61·18-s − 19-s + 1.27·20-s + 1.43·21-s + 2.91·22-s − 2.74·23-s − 2.75·24-s − 3.37·25-s + 1.67·26-s + 4.45·27-s + 0.521·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.59·3-s + 0.5·4-s + 0.570·5-s − 1.12·6-s + 0.197·7-s − 0.353·8-s + 1.53·9-s − 0.403·10-s − 0.879·11-s + 0.796·12-s − 0.464·13-s − 0.139·14-s + 0.908·15-s + 0.250·16-s − 0.342·17-s − 1.08·18-s − 0.229·19-s + 0.285·20-s + 0.313·21-s + 0.622·22-s − 0.573·23-s − 0.563·24-s − 0.674·25-s + 0.328·26-s + 0.857·27-s + 0.0985·28-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 \( 1 - 2.75T + 3T^{2} \)
5 \( 1 - 1.27T + 5T^{2} \)
7 \( 1 - 0.521T + 7T^{2} \)
11 \( 1 + 2.91T + 11T^{2} \)
13 \( 1 + 1.67T + 13T^{2} \)
17 \( 1 + 1.41T + 17T^{2} \)
23 \( 1 + 2.74T + 23T^{2} \)
29 \( 1 + 7.50T + 29T^{2} \)
31 \( 1 + 3.88T + 31T^{2} \)
37 \( 1 + 4.76T + 37T^{2} \)
41 \( 1 + 2.02T + 41T^{2} \)
43 \( 1 - 6.88T + 43T^{2} \)
47 \( 1 - 8.10T + 47T^{2} \)
53 \( 1 + 0.605T + 53T^{2} \)
59 \( 1 + 12.3T + 59T^{2} \)
61 \( 1 + 14.2T + 61T^{2} \)
67 \( 1 + 1.50T + 67T^{2} \)
71 \( 1 - 5.84T + 71T^{2} \)
73 \( 1 + 5.63T + 73T^{2} \)
79 \( 1 - 9.27T + 79T^{2} \)
83 \( 1 + 1.38T + 83T^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 - 7.33T + 97T^{2} \)
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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

−7.59832358332294968434194119520, −7.30138445663817772100118166407, −6.19528248190756385565869076194, −5.50696996100411837558239109053, −4.51813962030923096642929483454, −3.63413864099600864677280597721, −2.86789285801688233326948374340, −2.09208298905948603940566301683, −1.72588905494489394509277718328, 0, 1.72588905494489394509277718328, 2.09208298905948603940566301683, 2.86789285801688233326948374340, 3.63413864099600864677280597721, 4.51813962030923096642929483454, 5.50696996100411837558239109053, 6.19528248190756385565869076194, 7.30138445663817772100118166407, 7.59832358332294968434194119520

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