Properties

Degree 2
Conductor $ 3 \cdot 7 $
Sign $0.989 - 0.142i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 1.73i·3-s − 3·4-s − 6.92i·5-s + 1.73i·6-s + (1 + 6.92i)7-s − 7·8-s − 2.99·9-s − 6.92i·10-s + 10·11-s − 5.19i·12-s + 6.92i·13-s + (1 + 6.92i)14-s + 11.9·15-s + 5·16-s + ⋯
L(s)  = 1  + 0.5·2-s + 0.577i·3-s − 0.750·4-s − 1.38i·5-s + 0.288i·6-s + (0.142 + 0.989i)7-s − 0.875·8-s − 0.333·9-s − 0.692i·10-s + 0.909·11-s − 0.433i·12-s + 0.532i·13-s + (0.0714 + 0.494i)14-s + 0.799·15-s + 0.312·16-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.142i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.989 - 0.142i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(21\)    =    \(3 \cdot 7\)
\( \varepsilon \)  =  $0.989 - 0.142i$
motivic weight  =  \(2\)
character  :  $\chi_{21} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 21,\ (\ :1),\ 0.989 - 0.142i)$
$L(\frac{3}{2})$  $\approx$  $0.952768 + 0.0684057i$
$L(\frac12)$  $\approx$  $0.952768 + 0.0684057i$
$L(2)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - 1.73iT \)
7 \( 1 + (-1 - 6.92i)T \)
good2 \( 1 - T + 4T^{2} \)
5 \( 1 + 6.92iT - 25T^{2} \)
11 \( 1 - 10T + 121T^{2} \)
13 \( 1 - 6.92iT - 169T^{2} \)
17 \( 1 - 289T^{2} \)
19 \( 1 + 20.7iT - 361T^{2} \)
23 \( 1 + 14T + 529T^{2} \)
29 \( 1 + 38T + 841T^{2} \)
31 \( 1 - 27.7iT - 961T^{2} \)
37 \( 1 - 26T + 1.36e3T^{2} \)
41 \( 1 + 69.2iT - 1.68e3T^{2} \)
43 \( 1 - 26T + 1.84e3T^{2} \)
47 \( 1 - 27.7iT - 2.20e3T^{2} \)
53 \( 1 - 10T + 2.80e3T^{2} \)
59 \( 1 - 76.2iT - 3.48e3T^{2} \)
61 \( 1 - 34.6iT - 3.72e3T^{2} \)
67 \( 1 - 74T + 4.48e3T^{2} \)
71 \( 1 + 62T + 5.04e3T^{2} \)
73 \( 1 + 41.5iT - 5.32e3T^{2} \)
79 \( 1 + 46T + 6.24e3T^{2} \)
83 \( 1 - 90.0iT - 6.88e3T^{2} \)
89 \( 1 + 41.5iT - 7.92e3T^{2} \)
97 \( 1 - 55.4iT - 9.40e3T^{2} \)
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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

−17.81695983604413400532376722810, −16.68516973657824329419909365056, −15.41901003723195167156855700492, −14.14940590373736655263465816724, −12.77709226594676102208899896857, −11.76131059760542062142366906094, −9.278044820705301326040810468881, −8.795882945302853429926271393819, −5.56031607780747393425907988408, −4.31814990297491096689755315369, 3.71728929475028991669282655984, 6.22659888357279980443063344171, 7.76342273456707659248606363139, 9.904220807212029954536797316677, 11.40828011099156541309646669122, 13.01727907733575660680791099605, 14.19899250884021282252848878322, 14.75742959313816807996059400352, 16.99709269874029433437998324888, 18.10517069117350486220117120460

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