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 − 1.91·3-s + 4-s + 0.472·5-s + 1.91·6-s + 4.62·7-s − 8-s + 0.685·9-s − 0.472·10-s − 0.111·11-s − 1.91·12-s + 0.272·13-s − 4.62·14-s − 0.906·15-s + 16-s + 2.02·17-s − 0.685·18-s − 19-s + 0.472·20-s − 8.87·21-s + 0.111·22-s + 5.62·23-s + 1.91·24-s − 4.77·25-s − 0.272·26-s + 4.44·27-s + 4.62·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.10·3-s + 0.5·4-s + 0.211·5-s + 0.783·6-s + 1.74·7-s − 0.353·8-s + 0.228·9-s − 0.149·10-s − 0.0336·11-s − 0.554·12-s + 0.0755·13-s − 1.23·14-s − 0.234·15-s + 0.250·16-s + 0.491·17-s − 0.161·18-s − 0.229·19-s + 0.105·20-s − 1.93·21-s + 0.0237·22-s + 1.17·23-s + 0.391·24-s − 0.955·25-s − 0.0534·26-s + 0.855·27-s + 0.873·28-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 \( 1 + 1.91T + 3T^{2} \)
5 \( 1 - 0.472T + 5T^{2} \)
7 \( 1 - 4.62T + 7T^{2} \)
11 \( 1 + 0.111T + 11T^{2} \)
13 \( 1 - 0.272T + 13T^{2} \)
17 \( 1 - 2.02T + 17T^{2} \)
23 \( 1 - 5.62T + 23T^{2} \)
29 \( 1 + 4.88T + 29T^{2} \)
31 \( 1 + 8.39T + 31T^{2} \)
37 \( 1 - 7.54T + 37T^{2} \)
41 \( 1 + 5.51T + 41T^{2} \)
43 \( 1 - 0.252T + 43T^{2} \)
47 \( 1 - 3.79T + 47T^{2} \)
53 \( 1 + 2.47T + 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 + 7.76T + 61T^{2} \)
67 \( 1 + 13.7T + 67T^{2} \)
71 \( 1 - 7.29T + 71T^{2} \)
73 \( 1 + 9.98T + 73T^{2} \)
79 \( 1 + 8.19T + 79T^{2} \)
83 \( 1 - 4.34T + 83T^{2} \)
89 \( 1 - 1.85T + 89T^{2} \)
97 \( 1 + 4.37T + 97T^{2} \)
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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

−7.64635295903941796033844287721, −6.88928286729829410729074719640, −5.94459936021272518467146195667, −5.52539723951460306853395260128, −4.89750856608837836395080031195, −4.10956637021645258073755839216, −2.90176731607140057339792271018, −1.79460685637288267747373305326, −1.24057259133286688091771067493, 0, 1.24057259133286688091771067493, 1.79460685637288267747373305326, 2.90176731607140057339792271018, 4.10956637021645258073755839216, 4.89750856608837836395080031195, 5.52539723951460306853395260128, 5.94459936021272518467146195667, 6.88928286729829410729074719640, 7.64635295903941796033844287721

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