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  + 2·3-s − 4·5-s + 7-s + 9-s − 8·15-s − 2·17-s − 2·19-s + 2·21-s + 8·23-s + 11·25-s − 4·27-s + 2·29-s + 4·31-s − 4·35-s − 6·37-s − 2·41-s + 8·43-s − 4·45-s − 4·47-s + 49-s − 4·51-s − 10·53-s − 4·57-s + 6·59-s + 4·61-s + 63-s − 12·67-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.78·5-s + 0.377·7-s + 1/3·9-s − 2.06·15-s − 0.485·17-s − 0.458·19-s + 0.436·21-s + 1.66·23-s + 11/5·25-s − 0.769·27-s + 0.371·29-s + 0.718·31-s − 0.676·35-s − 0.986·37-s − 0.312·41-s + 1.21·43-s − 0.596·45-s − 0.583·47-s + 1/7·49-s − 0.560·51-s − 1.37·53-s − 0.529·57-s + 0.781·59-s + 0.512·61-s + 0.125·63-s − 1.46·67-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  =  $(2,\ 56,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9480191156$
$L(\frac12)$  $\approx$  $0.9480191156$
$L(\frac{3}{2})$   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 \( 1 - 2 T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−19.27268125804528, −19.08183222199248, −17.38432497732043, −15.97078376439103, −15.16780114648984, −14.47141648264652, −13.10747313979957, −11.85083361056234, −10.85093898417625, −8.928329697232845, −8.188057214509100, −7.144384347222307, −4.501855523711759, −3.141377929923908, 3.141377929923908, 4.501855523711759, 7.144384347222307, 8.188057214509100, 8.928329697232845, 10.85093898417625, 11.85083361056234, 13.10747313979957, 14.47141648264652, 15.16780114648984, 15.97078376439103, 17.38432497732043, 19.08183222199248, 19.27268125804528

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