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 − 2.56·3-s + 4-s + 3.62·5-s + 2.56·6-s + 4.81·7-s − 8-s + 3.58·9-s − 3.62·10-s + 0.483·11-s − 2.56·12-s − 5.40·13-s − 4.81·14-s − 9.29·15-s + 16-s − 5.64·17-s − 3.58·18-s − 19-s + 3.62·20-s − 12.3·21-s − 0.483·22-s − 3.29·23-s + 2.56·24-s + 8.13·25-s + 5.40·26-s − 1.48·27-s + 4.81·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.48·3-s + 0.5·4-s + 1.62·5-s + 1.04·6-s + 1.81·7-s − 0.353·8-s + 1.19·9-s − 1.14·10-s + 0.145·11-s − 0.740·12-s − 1.49·13-s − 1.28·14-s − 2.40·15-s + 0.250·16-s − 1.36·17-s − 0.843·18-s − 0.229·19-s + 0.810·20-s − 2.69·21-s − 0.103·22-s − 0.686·23-s + 0.523·24-s + 1.62·25-s + 1.06·26-s − 0.286·27-s + 0.909·28-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 \( 1 + 2.56T + 3T^{2} \)
5 \( 1 - 3.62T + 5T^{2} \)
7 \( 1 - 4.81T + 7T^{2} \)
11 \( 1 - 0.483T + 11T^{2} \)
13 \( 1 + 5.40T + 13T^{2} \)
17 \( 1 + 5.64T + 17T^{2} \)
23 \( 1 + 3.29T + 23T^{2} \)
29 \( 1 - 7.39T + 29T^{2} \)
31 \( 1 + 3.65T + 31T^{2} \)
37 \( 1 + 4.09T + 37T^{2} \)
41 \( 1 - 3.25T + 41T^{2} \)
43 \( 1 + 9.51T + 43T^{2} \)
47 \( 1 + 1.74T + 47T^{2} \)
53 \( 1 + 10.5T + 53T^{2} \)
59 \( 1 - 3.36T + 59T^{2} \)
61 \( 1 - 0.239T + 61T^{2} \)
67 \( 1 + 2.95T + 67T^{2} \)
71 \( 1 - 0.0139T + 71T^{2} \)
73 \( 1 - 8.52T + 73T^{2} \)
79 \( 1 - 4.43T + 79T^{2} \)
83 \( 1 + 3.63T + 83T^{2} \)
89 \( 1 + 13.1T + 89T^{2} \)
97 \( 1 + 7.06T + 97T^{2} \)
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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

−7.36098920286754513712057317125, −6.63592519228342542156191615825, −6.22513589817332268651084048176, −5.25702470170169728327703929529, −5.04977807445576186882689880914, −4.38637677099137181732397645065, −2.52717510352741671799054984578, −1.93134762340573115602053783819, −1.29913586826458149623850348836, 0, 1.29913586826458149623850348836, 1.93134762340573115602053783819, 2.52717510352741671799054984578, 4.38637677099137181732397645065, 5.04977807445576186882689880914, 5.25702470170169728327703929529, 6.22513589817332268651084048176, 6.63592519228342542156191615825, 7.36098920286754513712057317125

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