Properties

Degree 2
Conductor $ 2^{3} \cdot 7 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 7-s − 8-s − 9-s + 14-s + 16-s + 18-s + 2·23-s − 25-s − 28-s − 32-s − 36-s − 2·46-s + 49-s + 50-s + 56-s + 63-s + 64-s − 2·71-s + 72-s − 2·79-s + 81-s + 2·92-s − 98-s − 100-s − 112-s + ⋯
L(s)  = 1  − 2-s + 4-s − 7-s − 8-s − 9-s + 14-s + 16-s + 18-s + 2·23-s − 25-s − 28-s − 32-s − 36-s − 2·46-s + 49-s + 50-s + 56-s + 63-s + 64-s − 2·71-s + 72-s − 2·79-s + 81-s + 2·92-s − 98-s − 100-s − 112-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(56\)    =    \(2^{3} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{56} (13, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 56,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.3043447492\)
\(L(\frac12)\)  \(\approx\)  \(0.3043447492\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 + T \)
good3 \( 1 + T^{2} \)
5 \( 1 + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T^{2} \)
23 \( ( 1 - T )^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 + T )^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 + T )^{2} \)
83 \( 1 + T^{2} \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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

−15.79565908061006327148787841771, −14.73671678134309240431347460908, −13.20645456228466406130518818003, −11.90385973789881314134860691333, −10.82592427374314981434862061529, −9.573887196327588428321274714476, −8.622836713107787057397793052079, −7.13558216504220631858842167307, −5.85468181397150007828472600291, −3.00942554599034889633095542267, 3.00942554599034889633095542267, 5.85468181397150007828472600291, 7.13558216504220631858842167307, 8.622836713107787057397793052079, 9.573887196327588428321274714476, 10.82592427374314981434862061529, 11.90385973789881314134860691333, 13.20645456228466406130518818003, 14.73671678134309240431347460908, 15.79565908061006327148787841771

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