Properties

Degree 2
Conductor $ 3 \cdot 7 $
Sign $0.274 + 0.961i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.50i·2-s + (0.822 + 2.88i)3-s − 8.29·4-s + 1.24i·5-s + (10.1 − 2.88i)6-s + 2.64·7-s + 15.0i·8-s + (−7.64 + 4.74i)9-s + 4.35·10-s − 7.01i·11-s + (−6.82 − 23.9i)12-s − 11.6·13-s − 9.27i·14-s + (−3.58 + 1.02i)15-s + 19.5·16-s + 4.52i·17-s + ⋯
L(s)  = 1  − 1.75i·2-s + (0.274 + 0.961i)3-s − 2.07·4-s + 0.248i·5-s + (1.68 − 0.480i)6-s + 0.377·7-s + 1.88i·8-s + (−0.849 + 0.527i)9-s + 0.435·10-s − 0.637i·11-s + (−0.568 − 1.99i)12-s − 0.895·13-s − 0.662i·14-s + (−0.238 + 0.0681i)15-s + 1.22·16-s + 0.266i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.274 + 0.961i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.274 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(21\)    =    \(3 \cdot 7\)
\( \varepsilon \)  =  $0.274 + 0.961i$
motivic weight  =  \(2\)
character  :  $\chi_{21} (8, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 21,\ (\ :1),\ 0.274 + 0.961i)$
$L(\frac{3}{2})$  $\approx$  $0.689372 - 0.520236i$
$L(\frac12)$  $\approx$  $0.689372 - 0.520236i$
$L(2)$   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 \( 1 + (-0.822 - 2.88i)T \)
7 \( 1 - 2.64T \)
good2 \( 1 + 3.50iT - 4T^{2} \)
5 \( 1 - 1.24iT - 25T^{2} \)
11 \( 1 + 7.01iT - 121T^{2} \)
13 \( 1 + 11.6T + 169T^{2} \)
17 \( 1 - 4.52iT - 289T^{2} \)
19 \( 1 - 16.2T + 361T^{2} \)
23 \( 1 + 25.5iT - 529T^{2} \)
29 \( 1 - 9.49iT - 841T^{2} \)
31 \( 1 - 28.7T + 961T^{2} \)
37 \( 1 + 33.0T + 1.36e3T^{2} \)
41 \( 1 - 67.1iT - 1.68e3T^{2} \)
43 \( 1 + 24.1T + 1.84e3T^{2} \)
47 \( 1 + 33.0iT - 2.20e3T^{2} \)
53 \( 1 + 15.1iT - 2.80e3T^{2} \)
59 \( 1 - 92.3iT - 3.48e3T^{2} \)
61 \( 1 + 57.5T + 3.72e3T^{2} \)
67 \( 1 - 15.1T + 4.48e3T^{2} \)
71 \( 1 + 70.5iT - 5.04e3T^{2} \)
73 \( 1 + 76.7T + 5.32e3T^{2} \)
79 \( 1 - 127.T + 6.24e3T^{2} \)
83 \( 1 - 74.2iT - 6.88e3T^{2} \)
89 \( 1 + 127. iT - 7.92e3T^{2} \)
97 \( 1 + 23.1T + 9.40e3T^{2} \)
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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

−18.15500517166482548968659629605, −16.64744281187861099499080210241, −14.80496310009310581025844484908, −13.70589999221106094097002440358, −12.05215955094747283959721173692, −10.89543734001262305707354309184, −9.951468482733200695346432232765, −8.602616431874748734043358950999, −4.76339774791315908076859192989, −2.99522710624722766591794103112, 5.21070435478957792899505003238, 6.94700266496420815955978361283, 7.890427583737138323944897633325, 9.295502806591169357036349258163, 12.15326923744696763363963554493, 13.57754343007456276063292812852, 14.50827153543428751602231781918, 15.62957675556312300187269257014, 17.16241221816760159063219866595, 17.74689132243519088773079108945

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