Properties

Degree 2
Conductor 389
Sign $1$
Self-dual yes
Motivic weight 1

Related objects

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Normalization:  

Dirichlet series

L(E,s)  = 1  − 2·2-s − 2·3-s + 2·4-s − 3·5-s + 4·6-s − 5·7-s + 9-s + 6·10-s − 4·11-s − 4·12-s − 3·13-s + 10·14-s + 6·15-s − 4·16-s − 6·17-s − 2·18-s + 5·19-s − 6·20-s + 10·21-s + 8·22-s − 4·23-s + 4·25-s + 6·26-s + 4·27-s − 10·28-s − 6·29-s − 12·30-s + ⋯
L(s,E)  = 1  − 1.414·2-s − 1.154·3-s + 4-s − 1.341·5-s + 1.632·6-s − 1.889·7-s + 0.333·9-s + 1.897·10-s − 1.206·11-s − 1.154·12-s − 0.832·13-s + 2.672·14-s + 1.549·15-s − 16-s − 1.455·17-s − 0.471·18-s + 1.147·19-s − 1.341·20-s + 2.182·21-s + 1.705·22-s − 0.834·23-s + 0.8·25-s + 1.176·26-s + 0.769·27-s − 1.889·28-s − 1.114·29-s − 2.190·30-s + ⋯

Functional equation

\[\begin{align} \Lambda(E,s)=\mathstrut & 389 ^{s/2} \Gamma_{\C}(s) \cdot L(E,s)\cr =\mathstrut & \Lambda(E, 2-s) \end{align} \]
\[\begin{align} \Lambda(s,E)=\mathstrut & 389 ^{s/2} \Gamma_{\C}(s+1/2) \cdot L(s,E)\cr =\mathstrut & \Lambda(1-s,E) \end{align} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(389\)
\( \varepsilon \)  =  $1$
weight  =  1
Sato-Tate  :  SU(2)
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(2,\ 389,\ (\ :1/2),\ 1)$
$L(E,1)$  $=$  $0$
$L(\frac12,E)$  $=$  $0$
$L(E,\frac{3}{2})$   not available
$L(1,E)$   not available

Euler product

\[L(A,s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 389$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 389$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad389$1-T$
good2$1+2T+2T^{2}$
3$1+2T+3T^{2}$
5$1+3T+5T^{2}$
7$1+5T+7T^{2}$
11$1+4T+11T^{2}$
13$1+3T+13T^{2}$
17$1+6T+17T^{2}$
19$1-5T+19T^{2}$
23$1+4T+23T^{2}$
29$1+6T+29T^{2}$
31$1-4T+31T^{2}$
37$1+8T+37T^{2}$
41$1+3T+41T^{2}$
43$1-12T+43T^{2}$
47$1+2T+47T^{2}$
53$1+6T+53T^{2}$
59$1-3T+59T^{2}$
61$1+8T+61T^{2}$
67$1+5T+67T^{2}$
71$1+10T+71T^{2}$
73$1+7T+73T^{2}$
79$1+13T+79T^{2}$
83$1+12T+83T^{2}$
89$1+8T+89T^{2}$
97$1+9T+97T^{2}$
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\[\begin{equation} L(s,E) = \prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + p^{-2s})^{-1} \end{equation}\]

Imaginary part of the first few zeros on the critical line

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