Properties

Degree 2
Conductor $ 2 \cdot 19 \cdot 211 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2.28·3-s + 4-s + 4.33·5-s − 2.28·6-s − 0.955·7-s + 8-s + 2.21·9-s + 4.33·10-s − 5.13·11-s − 2.28·12-s − 0.223·13-s − 0.955·14-s − 9.89·15-s + 16-s + 2.89·17-s + 2.21·18-s − 19-s + 4.33·20-s + 2.18·21-s − 5.13·22-s − 3.30·23-s − 2.28·24-s + 13.7·25-s − 0.223·26-s + 1.79·27-s − 0.955·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.31·3-s + 0.5·4-s + 1.93·5-s − 0.932·6-s − 0.361·7-s + 0.353·8-s + 0.737·9-s + 1.37·10-s − 1.54·11-s − 0.659·12-s − 0.0620·13-s − 0.255·14-s − 2.55·15-s + 0.250·16-s + 0.701·17-s + 0.521·18-s − 0.229·19-s + 0.969·20-s + 0.476·21-s − 1.09·22-s − 0.689·23-s − 0.466·24-s + 2.75·25-s − 0.0438·26-s + 0.345·27-s − 0.180·28-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  =  0
Selberg data  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.485246021$
$L(\frac12)$  $\approx$  $2.485246021$
$L(\frac{3}{2})$   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 \( 1 + 2.28T + 3T^{2} \)
5 \( 1 - 4.33T + 5T^{2} \)
7 \( 1 + 0.955T + 7T^{2} \)
11 \( 1 + 5.13T + 11T^{2} \)
13 \( 1 + 0.223T + 13T^{2} \)
17 \( 1 - 2.89T + 17T^{2} \)
23 \( 1 + 3.30T + 23T^{2} \)
29 \( 1 + 0.0537T + 29T^{2} \)
31 \( 1 + 1.90T + 31T^{2} \)
37 \( 1 + 2.29T + 37T^{2} \)
41 \( 1 - 0.965T + 41T^{2} \)
43 \( 1 + 5.82T + 43T^{2} \)
47 \( 1 - 9.69T + 47T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 - 4.27T + 59T^{2} \)
61 \( 1 - 1.64T + 61T^{2} \)
67 \( 1 + 11.7T + 67T^{2} \)
71 \( 1 - 6.06T + 71T^{2} \)
73 \( 1 - 3.78T + 73T^{2} \)
79 \( 1 - 9.19T + 79T^{2} \)
83 \( 1 - 16.2T + 83T^{2} \)
89 \( 1 - 15.5T + 89T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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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

−7.52774232229779933191738949797, −6.76848140981102882361467737125, −6.09062941735863093495104293202, −5.77756110186190877756967423654, −5.16611045465621760986757576199, −4.82900030021281993968793221236, −3.47718508368481767651256621509, −2.53760150489885000788331735568, −1.94732994441215179260122903196, −0.73290722076026772522775902025, 0.73290722076026772522775902025, 1.94732994441215179260122903196, 2.53760150489885000788331735568, 3.47718508368481767651256621509, 4.82900030021281993968793221236, 5.16611045465621760986757576199, 5.77756110186190877756967423654, 6.09062941735863093495104293202, 6.76848140981102882361467737125, 7.52774232229779933191738949797

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