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 − 1.10·3-s + 4-s + 3.90·5-s − 1.10·6-s − 4.77·7-s + 8-s − 1.76·9-s + 3.90·10-s + 5.69·11-s − 1.10·12-s − 4.09·13-s − 4.77·14-s − 4.32·15-s + 16-s − 0.265·17-s − 1.76·18-s − 19-s + 3.90·20-s + 5.30·21-s + 5.69·22-s − 3.37·23-s − 1.10·24-s + 10.2·25-s − 4.09·26-s + 5.29·27-s − 4.77·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.640·3-s + 0.5·4-s + 1.74·5-s − 0.453·6-s − 1.80·7-s + 0.353·8-s − 0.589·9-s + 1.23·10-s + 1.71·11-s − 0.320·12-s − 1.13·13-s − 1.27·14-s − 1.11·15-s + 0.250·16-s − 0.0642·17-s − 0.416·18-s − 0.229·19-s + 0.872·20-s + 1.15·21-s + 1.21·22-s − 0.703·23-s − 0.226·24-s + 2.04·25-s − 0.802·26-s + 1.01·27-s − 0.902·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.690022082$
$L(\frac12)$  $\approx$  $2.690022082$
$L(\frac{3}{2})$   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 \( 1 + 1.10T + 3T^{2} \)
5 \( 1 - 3.90T + 5T^{2} \)
7 \( 1 + 4.77T + 7T^{2} \)
11 \( 1 - 5.69T + 11T^{2} \)
13 \( 1 + 4.09T + 13T^{2} \)
17 \( 1 + 0.265T + 17T^{2} \)
23 \( 1 + 3.37T + 23T^{2} \)
29 \( 1 + 8.30T + 29T^{2} \)
31 \( 1 + 6.89T + 31T^{2} \)
37 \( 1 - 10.1T + 37T^{2} \)
41 \( 1 - 10.0T + 41T^{2} \)
43 \( 1 + 2.35T + 43T^{2} \)
47 \( 1 - 0.906T + 47T^{2} \)
53 \( 1 - 11.1T + 53T^{2} \)
59 \( 1 - 14.5T + 59T^{2} \)
61 \( 1 - 3.09T + 61T^{2} \)
67 \( 1 - 14.6T + 67T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + 4.58T + 73T^{2} \)
79 \( 1 + 5.66T + 79T^{2} \)
83 \( 1 - 9.13T + 83T^{2} \)
89 \( 1 + 6.19T + 89T^{2} \)
97 \( 1 - 8.89T + 97T^{2} \)
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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

−7.36606572991955156009990062849, −6.76659905829728764914327725996, −6.25342536768827627440432988850, −5.78410274165270585251017738397, −5.44265068628264663332667527781, −4.25864328314813247567229568215, −3.56690435135525439809899062188, −2.58638612647275465261959401816, −2.06804289486049022147777561890, −0.72665901864605275519388150901, 0.72665901864605275519388150901, 2.06804289486049022147777561890, 2.58638612647275465261959401816, 3.56690435135525439809899062188, 4.25864328314813247567229568215, 5.44265068628264663332667527781, 5.78410274165270585251017738397, 6.25342536768827627440432988850, 6.76659905829728764914327725996, 7.36606572991955156009990062849

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