Properties

Degree 2
Conductor $ 2^{5} \cdot 431 $
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  + 3·3-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

\[\begin{aligned} \Lambda(s)=\mathstrut & 13792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 13792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(13792\)    =    \(2^{5} \cdot 431\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{13792} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 13792,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.102508280$
$L(\frac12)$  $\approx$  $6.102508280$
$L(\frac{3}{2})$   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 \( 1 + 5 T + p T^{2} \)
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

−16.26545554660951, −15.60216638748858, −14.81557276184326, −14.50035443164567, −13.67862470310532, −13.47575700653597, −13.24514570329200, −12.54656765893107, −11.68209310473528, −10.92278112362572, −10.04376761206625, −9.641375138637432, −9.167232559319697, −8.989811998308886, −8.304771401885276, −7.202493360538519, −6.823148323811517, −6.238486737563783, −5.652691849716780, −4.408327369801167, −3.799177784756520, −3.126761082265812, −2.671511842703536, −1.738336305129901, −1.097395395212680, 1.097395395212680, 1.738336305129901, 2.671511842703536, 3.126761082265812, 3.799177784756520, 4.408327369801167, 5.652691849716780, 6.238486737563783, 6.823148323811517, 7.202493360538519, 8.304771401885276, 8.989811998308886, 9.167232559319697, 9.641375138637432, 10.04376761206625, 10.92278112362572, 11.68209310473528, 12.54656765893107, 13.24514570329200, 13.47575700653597, 13.67862470310532, 14.50035443164567, 14.81557276184326, 15.60216638748858, 16.26545554660951

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