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  − 2·2-s + 3-s + 2·4-s − 5-s − 2·6-s + 7-s + 9-s + 2·10-s + 5·11-s + 2·12-s − 4·13-s − 2·14-s − 15-s − 4·16-s + 6·17-s − 2·18-s + 5·19-s − 2·20-s + 21-s − 10·22-s + 25-s + 8·26-s + 27-s + 2·28-s + 6·29-s + 2·30-s + 5·31-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s + 0.377·7-s + 1/3·9-s + 0.632·10-s + 1.50·11-s + 0.577·12-s − 1.10·13-s − 0.534·14-s − 0.258·15-s − 16-s + 1.45·17-s − 0.471·18-s + 1.14·19-s − 0.447·20-s + 0.218·21-s − 2.13·22-s + 1/5·25-s + 1.56·26-s + 0.192·27-s + 0.377·28-s + 1.11·29-s + 0.365·30-s + 0.898·31-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  =  $(2,\ 55545,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.925977887$
$L(\frac12)$  $\approx$  $1.925977887$
$L(\frac{3}{2})$   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 \( 1 + p T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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

−14.44682325268079, −14.05216258195612, −13.61323386799821, −12.66674765484365, −12.15352225309323, −11.75106861481597, −11.38579775427712, −10.60758104862282, −9.978824734817689, −9.721875787136984, −9.275539788171180, −8.714563632800583, −8.144065227690843, −7.723288819319192, −7.343671168986370, −6.802869698291406, −6.127369125815496, −5.281563292315783, −4.485308808093802, −4.198214845735454, −3.154690929478710, −2.795558464759269, −1.780063433854854, −1.167170245824922, −0.6969034426565653, 0.6969034426565653, 1.167170245824922, 1.780063433854854, 2.795558464759269, 3.154690929478710, 4.198214845735454, 4.485308808093802, 5.281563292315783, 6.127369125815496, 6.802869698291406, 7.343671168986370, 7.723288819319192, 8.144065227690843, 8.714563632800583, 9.275539788171180, 9.721875787136984, 9.978824734817689, 10.60758104862282, 11.38579775427712, 11.75106861481597, 12.15352225309323, 12.66674765484365, 13.61323386799821, 14.05216258195612, 14.44682325268079

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