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 + 0.812·3-s + 4-s − 3.14·5-s + 0.812·6-s + 3.94·7-s + 8-s − 2.34·9-s − 3.14·10-s + 5.89·11-s + 0.812·12-s − 4.62·13-s + 3.94·14-s − 2.55·15-s + 16-s + 3.11·17-s − 2.34·18-s − 19-s − 3.14·20-s + 3.20·21-s + 5.89·22-s + 0.559·23-s + 0.812·24-s + 4.90·25-s − 4.62·26-s − 4.33·27-s + 3.94·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.468·3-s + 0.5·4-s − 1.40·5-s + 0.331·6-s + 1.49·7-s + 0.353·8-s − 0.780·9-s − 0.995·10-s + 1.77·11-s + 0.234·12-s − 1.28·13-s + 1.05·14-s − 0.659·15-s + 0.250·16-s + 0.755·17-s − 0.551·18-s − 0.229·19-s − 0.703·20-s + 0.699·21-s + 1.25·22-s + 0.116·23-s + 0.165·24-s + 0.980·25-s − 0.906·26-s − 0.834·27-s + 0.745·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.697916760$
$L(\frac12)$  $\approx$  $3.697916760$
$L(\frac{3}{2})$   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 \( 1 - 0.812T + 3T^{2} \)
5 \( 1 + 3.14T + 5T^{2} \)
7 \( 1 - 3.94T + 7T^{2} \)
11 \( 1 - 5.89T + 11T^{2} \)
13 \( 1 + 4.62T + 13T^{2} \)
17 \( 1 - 3.11T + 17T^{2} \)
23 \( 1 - 0.559T + 23T^{2} \)
29 \( 1 - 8.95T + 29T^{2} \)
31 \( 1 + 1.19T + 31T^{2} \)
37 \( 1 - 3.33T + 37T^{2} \)
41 \( 1 - 8.62T + 41T^{2} \)
43 \( 1 + 2.86T + 43T^{2} \)
47 \( 1 + 4.24T + 47T^{2} \)
53 \( 1 - 1.11T + 53T^{2} \)
59 \( 1 + 6.80T + 59T^{2} \)
61 \( 1 + 3.69T + 61T^{2} \)
67 \( 1 - 3.18T + 67T^{2} \)
71 \( 1 - 6.64T + 71T^{2} \)
73 \( 1 + 15.2T + 73T^{2} \)
79 \( 1 - 12.6T + 79T^{2} \)
83 \( 1 - 1.11T + 83T^{2} \)
89 \( 1 + 13.6T + 89T^{2} \)
97 \( 1 - 16.6T + 97T^{2} \)
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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

−7.81866026392651825095041685757, −7.30333510384173817961993203227, −6.49969700304546895807893412266, −5.62341382388803322002669227899, −4.61401246655895661027581307437, −4.47648082054619040297604669754, −3.58181797649039381976770867664, −2.89933278826189954660684441788, −1.91097063597500869174472793462, −0.860602869458505049597234639337, 0.860602869458505049597234639337, 1.91097063597500869174472793462, 2.89933278826189954660684441788, 3.58181797649039381976770867664, 4.47648082054619040297604669754, 4.61401246655895661027581307437, 5.62341382388803322002669227899, 6.49969700304546895807893412266, 7.30333510384173817961993203227, 7.81866026392651825095041685757

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