Properties

Degree 2
Conductor $ 2^{6} \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 + 7-s + 9-s + 4·13-s + 6·17-s − 2·19-s + 2·21-s − 5·25-s − 4·27-s + 6·29-s − 4·31-s − 2·37-s + 8·39-s + 6·41-s − 8·43-s − 12·47-s + 49-s + 12·51-s − 6·53-s − 4·57-s + 6·59-s − 8·61-s + 63-s + 4·67-s + 2·73-s − 10·75-s + 8·79-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.10·13-s + 1.45·17-s − 0.458·19-s + 0.436·21-s − 25-s − 0.769·27-s + 1.11·29-s − 0.718·31-s − 0.328·37-s + 1.28·39-s + 0.937·41-s − 1.21·43-s − 1.75·47-s + 1/7·49-s + 1.68·51-s − 0.824·53-s − 0.529·57-s + 0.781·59-s − 1.02·61-s + 0.125·63-s + 0.488·67-s + 0.234·73-s − 1.15·75-s + 0.900·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(448\)    =    \(2^{6} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{448} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 448,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.101530499$
$L(\frac12)$  $\approx$  $2.101530499$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−19.76599144195716, −19.29161593578283, −18.44850221831394, −17.72458504966021, −16.71184646215088, −15.92005196820820, −15.05195630081784, −14.37299136416993, −13.79554581641985, −12.99456737913635, −11.96768802253890, −11.06714116909225, −10.04491938039308, −9.195409247501139, −8.222985047096640, −7.870217720218150, −6.477813773859231, −5.366638023737943, −3.935221195750085, −3.073026859383028, −1.649505021347734, 1.649505021347734, 3.073026859383028, 3.935221195750085, 5.366638023737943, 6.477813773859231, 7.870217720218150, 8.222985047096640, 9.195409247501139, 10.04491938039308, 11.06714116909225, 11.96768802253890, 12.99456737913635, 13.79554581641985, 14.37299136416993, 15.05195630081784, 15.92005196820820, 16.71184646215088, 17.72458504966021, 18.44850221831394, 19.29161593578283, 19.76599144195716

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