Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 \cdot 23^{2} $
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 − 2·4-s − 5-s + 7-s + 9-s − 2·12-s − 13-s − 15-s + 4·16-s + 3·17-s + 2·19-s + 2·20-s + 21-s + 25-s + 27-s − 2·28-s − 3·29-s + 2·31-s − 35-s − 2·36-s − 4·37-s − 39-s + 12·41-s − 10·43-s − 45-s − 9·47-s + 4·48-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.577·12-s − 0.277·13-s − 0.258·15-s + 16-s + 0.727·17-s + 0.458·19-s + 0.447·20-s + 0.218·21-s + 1/5·25-s + 0.192·27-s − 0.377·28-s − 0.557·29-s + 0.359·31-s − 0.169·35-s − 1/3·36-s − 0.657·37-s − 0.160·39-s + 1.87·41-s − 1.52·43-s − 0.149·45-s − 1.31·47-s + 0.577·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(55545\)    =    \(3 \cdot 5 \cdot 7 \cdot 23^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{55545} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 55545,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.959931624$
$L(\frac12)$  $\approx$  $1.959931624$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7,\;23\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7,\;23\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
23 \( 1 \)
good2 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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

−14.39416010777458, −13.99758133741012, −13.42215557021230, −12.92544886133324, −12.48552222042461, −11.90721396703143, −11.42713246330034, −10.76226146941223, −10.01009672091494, −9.861582742630466, −9.114064026684462, −8.706560652249426, −8.188768651913278, −7.601894688940030, −7.396421643949463, −6.464692101759879, −5.769723619854418, −5.122914878765372, −4.693328372455770, −4.081351091259511, −3.432670615292956, −3.052271684847684, −2.060624821177096, −1.289784202054056, −0.5102843784970747, 0.5102843784970747, 1.289784202054056, 2.060624821177096, 3.052271684847684, 3.432670615292956, 4.081351091259511, 4.693328372455770, 5.122914878765372, 5.769723619854418, 6.464692101759879, 7.396421643949463, 7.601894688940030, 8.188768651913278, 8.706560652249426, 9.114064026684462, 9.861582742630466, 10.01009672091494, 10.76226146941223, 11.42713246330034, 11.90721396703143, 12.48552222042461, 12.92544886133324, 13.42215557021230, 13.99758133741012, 14.39416010777458

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