Properties

Degree 2
Conductor $ 2^{3} \cdot 47 \cdot 73 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 3·5-s − 4·7-s + 6·9-s − 4·11-s − 5·13-s + 9·15-s − 6·17-s − 2·19-s + 12·21-s − 8·23-s + 4·25-s − 9·27-s − 9·29-s + 4·31-s + 12·33-s + 12·35-s − 37-s + 15·39-s − 10·41-s − 5·43-s − 18·45-s − 47-s + 9·49-s + 18·51-s − 4·53-s + 12·55-s + ⋯
L(s)  = 1  − 1.73·3-s − 1.34·5-s − 1.51·7-s + 2·9-s − 1.20·11-s − 1.38·13-s + 2.32·15-s − 1.45·17-s − 0.458·19-s + 2.61·21-s − 1.66·23-s + 4/5·25-s − 1.73·27-s − 1.67·29-s + 0.718·31-s + 2.08·33-s + 2.02·35-s − 0.164·37-s + 2.40·39-s − 1.56·41-s − 0.762·43-s − 2.68·45-s − 0.145·47-s + 9/7·49-s + 2.52·51-s − 0.549·53-s + 1.61·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27448\)    =    \(2^{3} \cdot 47 \cdot 73\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{27448} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(2,\ 27448,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;47,\;73\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;47,\;73\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
47 \( 1 + T \)
73 \( 1 - T \)
good3 \( 1 + p T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 15 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
79 \( 1 + 17 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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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

−15.90930776747648, −15.71369235301292, −15.36253995719904, −14.75616135335327, −13.50345737083202, −13.29149899939537, −12.57821744945434, −12.19947195594755, −11.94074569207676, −11.18771797928288, −10.80604103966448, −10.15765412806717, −9.835131164102358, −9.126087139964090, −8.112274837352586, −7.712105676326037, −6.939104575792801, −6.704767546250983, −5.998337598644545, −5.406974334195293, −4.673583830524166, −4.294220509869643, −3.544410766732975, −2.720451563689213, −1.821966159135146, 0, 0, 0, 1.821966159135146, 2.720451563689213, 3.544410766732975, 4.294220509869643, 4.673583830524166, 5.406974334195293, 5.998337598644545, 6.704767546250983, 6.939104575792801, 7.712105676326037, 8.112274837352586, 9.126087139964090, 9.835131164102358, 10.15765412806717, 10.80604103966448, 11.18771797928288, 11.94074569207676, 12.19947195594755, 12.57821744945434, 13.29149899939537, 13.50345737083202, 14.75616135335327, 15.36253995719904, 15.71369235301292, 15.90930776747648

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