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 + 0.731·3-s + 4-s + 3.24·5-s + 0.731·6-s + 4.02·7-s + 8-s − 2.46·9-s + 3.24·10-s + 1.64·11-s + 0.731·12-s + 2.18·13-s + 4.02·14-s + 2.37·15-s + 16-s − 2.11·17-s − 2.46·18-s − 19-s + 3.24·20-s + 2.94·21-s + 1.64·22-s + 7.93·23-s + 0.731·24-s + 5.50·25-s + 2.18·26-s − 3.99·27-s + 4.02·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.422·3-s + 0.5·4-s + 1.44·5-s + 0.298·6-s + 1.52·7-s + 0.353·8-s − 0.821·9-s + 1.02·10-s + 0.495·11-s + 0.211·12-s + 0.605·13-s + 1.07·14-s + 0.612·15-s + 0.250·16-s − 0.513·17-s − 0.581·18-s − 0.229·19-s + 0.724·20-s + 0.642·21-s + 0.350·22-s + 1.65·23-s + 0.149·24-s + 1.10·25-s + 0.428·26-s − 0.769·27-s + 0.760·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.463126862$
$L(\frac12)$  $\approx$  $6.463126862$
$L(\frac{3}{2})$   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 \( 1 - 0.731T + 3T^{2} \)
5 \( 1 - 3.24T + 5T^{2} \)
7 \( 1 - 4.02T + 7T^{2} \)
11 \( 1 - 1.64T + 11T^{2} \)
13 \( 1 - 2.18T + 13T^{2} \)
17 \( 1 + 2.11T + 17T^{2} \)
23 \( 1 - 7.93T + 23T^{2} \)
29 \( 1 + 7.04T + 29T^{2} \)
31 \( 1 - 2.96T + 31T^{2} \)
37 \( 1 + 1.09T + 37T^{2} \)
41 \( 1 - 5.86T + 41T^{2} \)
43 \( 1 + 10.0T + 43T^{2} \)
47 \( 1 - 7.44T + 47T^{2} \)
53 \( 1 + 1.80T + 53T^{2} \)
59 \( 1 + 8.17T + 59T^{2} \)
61 \( 1 + 9.55T + 61T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 - 2.43T + 71T^{2} \)
73 \( 1 - 0.129T + 73T^{2} \)
79 \( 1 - 2.09T + 79T^{2} \)
83 \( 1 + 1.41T + 83T^{2} \)
89 \( 1 + 7.58T + 89T^{2} \)
97 \( 1 - 1.83T + 97T^{2} \)
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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

−7.84660463721182582191571588155, −7.01563963535246591940072472735, −6.25544118944319891227482188269, −5.66058417693321389614771362873, −5.10578480488433577472634518266, −4.44035295189510568599647423975, −3.45611560333857422259846778930, −2.58508655660716549296240168194, −1.90679946457608528662640672253, −1.25413804884885295908263116030, 1.25413804884885295908263116030, 1.90679946457608528662640672253, 2.58508655660716549296240168194, 3.45611560333857422259846778930, 4.44035295189510568599647423975, 5.10578480488433577472634518266, 5.66058417693321389614771362873, 6.25544118944319891227482188269, 7.01563963535246591940072472735, 7.84660463721182582191571588155

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