Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 7-s − 3·9-s − 4·11-s + 2·13-s − 6·17-s + 8·19-s − 25-s + 6·29-s + 8·31-s − 2·35-s − 2·37-s + 2·41-s − 4·43-s − 6·45-s − 8·47-s + 49-s + 6·53-s − 8·55-s − 6·61-s + 3·63-s + 4·65-s − 4·67-s − 8·71-s + 10·73-s + 4·77-s + 16·79-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.377·7-s − 9-s − 1.20·11-s + 0.554·13-s − 1.45·17-s + 1.83·19-s − 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.338·35-s − 0.328·37-s + 0.312·41-s − 0.609·43-s − 0.894·45-s − 1.16·47-s + 1/7·49-s + 0.824·53-s − 1.07·55-s − 0.768·61-s + 0.377·63-s + 0.496·65-s − 0.488·67-s − 0.949·71-s + 1.17·73-s + 0.455·77-s + 1.80·79-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(56\)    =    \(2^{3} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{56} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 56,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8745483141$
$L(\frac12)$  $\approx$  $0.8745483141$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
7 \( 1 + T \)
good3 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.55018827324056, −18.07433600423375, −17.68571678800724, −16.28523153907899, −15.41323382474483, −13.77746713200327, −13.45094853660694, −11.84146297682029, −10.60224244605800, −9.457349313423188, −8.188420417684288, −6.422620848135640, −5.223164094353384, −2.791838006127257, 2.791838006127257, 5.223164094353384, 6.422620848135640, 8.188420417684288, 9.457349313423188, 10.60224244605800, 11.84146297682029, 13.45094853660694, 13.77746713200327, 15.41323382474483, 16.28523153907899, 17.68571678800724, 18.07433600423375, 19.55018827324056

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