Properties

Degree 2
Conductor $ 2^{6} \cdot 7 $
Sign $0.755 - 0.654i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 3.46i·5-s + (2 − 1.73i)7-s + 9-s + 3.46i·11-s − 3.46i·13-s + 6.92i·15-s + 2·19-s + (4 − 3.46i)21-s + 3.46i·23-s − 6.99·25-s − 4·27-s − 6·29-s + 8·31-s + 6.92i·33-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.54i·5-s + (0.755 − 0.654i)7-s + 0.333·9-s + 1.04i·11-s − 0.960i·13-s + 1.78i·15-s + 0.458·19-s + (0.872 − 0.755i)21-s + 0.722i·23-s − 1.39·25-s − 0.769·27-s − 1.11·29-s + 1.43·31-s + 1.20i·33-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(448\)    =    \(2^{6} \cdot 7\)
\( \varepsilon \)  =  $0.755 - 0.654i$
motivic weight  =  \(1\)
character  :  $\chi_{448} (447, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 448,\ (\ :1/2),\ 0.755 - 0.654i)$
$L(1)$  $\approx$  $1.95274 + 0.728030i$
$L(\frac12)$  $\approx$  $1.95274 + 0.728030i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7\}$, \(F_p(T)\) 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 \( 1 \)
7 \( 1 + (-2 + 1.73i)T \)
good3 \( 1 - 2T + 3T^{2} \)
5 \( 1 - 3.46iT - 5T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 - 3.46iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 6.92iT - 41T^{2} \)
43 \( 1 + 10.3iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 + 3.46iT - 61T^{2} \)
67 \( 1 - 3.46iT - 67T^{2} \)
71 \( 1 + 3.46iT - 71T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 + 3.46iT - 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + 6.92iT - 89T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−11.01287155568625878927977115204, −10.26292381518721065546968140955, −9.550059819250600182882265266760, −8.247623890924368214748670444184, −7.52496880825596917724798965123, −6.98708042166248761297055838228, −5.50244752816487965820653538501, −3.99748740106848241050424196853, −3.08433232669730184582890431382, −2.05462471309512005043083227882, 1.40351027887794056703929038807, 2.73617141151098625327292214363, 4.17067380789998926272961730796, 5.08387460991897900681181111709, 6.16478904905507191535804434872, 7.907070679167903386448456167174, 8.345910353654609925558497678554, 9.036128236832154519105890323137, 9.588953830211663260248387217318, 11.25706804491268867726777168842

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