Properties

Degree 2
Conductor $ 2^{3} \cdot 47 \cdot 73 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 4·7-s − 2·9-s + 4·11-s − 5·13-s + 15-s + 6·17-s + 2·19-s + 4·21-s − 2·23-s − 4·25-s − 5·27-s + 3·29-s + 8·31-s + 4·33-s + 4·35-s + 7·37-s − 5·39-s − 5·43-s − 2·45-s − 47-s + 9·49-s + 6·51-s + 6·53-s + 4·55-s + 2·57-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1.51·7-s − 2/3·9-s + 1.20·11-s − 1.38·13-s + 0.258·15-s + 1.45·17-s + 0.458·19-s + 0.872·21-s − 0.417·23-s − 4/5·25-s − 0.962·27-s + 0.557·29-s + 1.43·31-s + 0.696·33-s + 0.676·35-s + 1.15·37-s − 0.800·39-s − 0.762·43-s − 0.298·45-s − 0.145·47-s + 9/7·49-s + 0.840·51-s + 0.824·53-s + 0.539·55-s + 0.264·57-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  =  0
Selberg data  =  $(2,\ 27448,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.328929742$
$L(\frac12)$  $\approx$  $4.328929742$
$L(\frac{3}{2})$   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 \( 1 - T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
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 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.80241055144665, −14.75389344401906, −14.20694608006397, −13.92994154064078, −13.37994875195133, −12.30045893502677, −12.03482375233988, −11.55627776283996, −11.14918208668762, −10.10566923909034, −9.776629844314666, −9.403155203743375, −8.400694222032322, −8.223654403677700, −7.698836562368268, −7.000292760080818, −6.286155612738182, −5.428786548819190, −5.237392606564596, −4.329144607750445, −3.798142560962253, −2.832545213756898, −2.325123583997196, −1.551862851288236, −0.8262727231579806, 0.8262727231579806, 1.551862851288236, 2.325123583997196, 2.832545213756898, 3.798142560962253, 4.329144607750445, 5.237392606564596, 5.428786548819190, 6.286155612738182, 7.000292760080818, 7.698836562368268, 8.223654403677700, 8.400694222032322, 9.403155203743375, 9.776629844314666, 10.10566923909034, 11.14918208668762, 11.55627776283996, 12.03482375233988, 12.30045893502677, 13.37994875195133, 13.92994154064078, 14.20694608006397, 14.75389344401906, 14.80241055144665

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