Properties

Degree 2
Conductor $ 2^{2} \cdot 19 $
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  + 2·3-s − 5-s − 3·7-s + 9-s + 5·11-s − 4·13-s − 2·15-s − 3·17-s − 19-s − 6·21-s + 8·23-s − 4·25-s − 4·27-s − 2·29-s + 4·31-s + 10·33-s + 3·35-s + 10·37-s − 8·39-s + 10·41-s + 43-s − 45-s − 47-s + 2·49-s − 6·51-s − 4·53-s − 5·55-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s + 1.50·11-s − 1.10·13-s − 0.516·15-s − 0.727·17-s − 0.229·19-s − 1.30·21-s + 1.66·23-s − 4/5·25-s − 0.769·27-s − 0.371·29-s + 0.718·31-s + 1.74·33-s + 0.507·35-s + 1.64·37-s − 1.28·39-s + 1.56·41-s + 0.152·43-s − 0.149·45-s − 0.145·47-s + 2/7·49-s − 0.840·51-s − 0.549·53-s − 0.674·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(76\)    =    \(2^{2} \cdot 19\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{76} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 76,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.110419746$
$L(\frac12)$  $\approx$  $1.110419746$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;19\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;19\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
19 \( 1 + T \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−19.52471042637900, −19.13356120455157, −17.40724029957344, −16.52353459774370, −15.24225419288510, −14.63931551152130, −13.52350522314312, −12.52817820768231, −11.31944737144510, −9.553328544165326, −9.084561409834522, −7.624292291523759, −6.443235140922679, −4.155372769559701, −2.825112610707033, 2.825112610707033, 4.155372769559701, 6.443235140922679, 7.624292291523759, 9.084561409834522, 9.553328544165326, 11.31944737144510, 12.52817820768231, 13.52350522314312, 14.63931551152130, 15.24225419288510, 16.52353459774370, 17.40724029957344, 19.13356120455157, 19.52471042637900

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