Properties

Degree 2
Conductor $ 2^{6} \cdot 431 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 3·5-s + 4·7-s + 6·9-s + 5·11-s − 4·13-s − 9·15-s − 6·17-s + 7·19-s + 12·21-s − 3·23-s + 4·25-s + 9·27-s − 3·29-s + 4·31-s + 15·33-s − 12·35-s − 8·37-s − 12·39-s − 6·41-s + 8·43-s − 18·45-s + 6·47-s + 9·49-s − 18·51-s − 53-s − 15·55-s + ⋯
L(s)  = 1  + 1.73·3-s − 1.34·5-s + 1.51·7-s + 2·9-s + 1.50·11-s − 1.10·13-s − 2.32·15-s − 1.45·17-s + 1.60·19-s + 2.61·21-s − 0.625·23-s + 4/5·25-s + 1.73·27-s − 0.557·29-s + 0.718·31-s + 2.61·33-s − 2.02·35-s − 1.31·37-s − 1.92·39-s − 0.937·41-s + 1.21·43-s − 2.68·45-s + 0.875·47-s + 9/7·49-s − 2.52·51-s − 0.137·53-s − 2.02·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 27584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 27584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27584\)    =    \(2^{6} \cdot 431\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{27584} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 27584,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.733567115$
$L(\frac12)$  $\approx$  $4.733567115$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;431\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;431\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
431 \( 1 + T \)
good3 \( 1 - p T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−15.19536465952152, −14.60595309956603, −14.36814983754102, −13.73987494335782, −13.51127985894314, −12.30654646971010, −12.08467230726738, −11.60251144221869, −11.07171872667483, −10.31214397327611, −9.522504025544569, −9.071729924437952, −8.665327341232488, −8.000611524633715, −7.730398508831507, −7.136416150567028, −6.762359804556164, −5.456529901320507, −4.610534619174335, −4.314368219192006, −3.726169886889051, −3.147042125035904, −2.191248818104190, −1.760663908179375, −0.7830351385025000, 0.7830351385025000, 1.760663908179375, 2.191248818104190, 3.147042125035904, 3.726169886889051, 4.314368219192006, 4.610534619174335, 5.456529901320507, 6.762359804556164, 7.136416150567028, 7.730398508831507, 8.000611524633715, 8.665327341232488, 9.071729924437952, 9.522504025544569, 10.31214397327611, 11.07171872667483, 11.60251144221869, 12.08467230726738, 12.30654646971010, 13.51127985894314, 13.73987494335782, 14.36814983754102, 14.60595309956603, 15.19536465952152

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