Properties

Degree 2
Conductor $ 3 \cdot 7 $
Sign $0.266 + 0.963i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.93 − 1.11i)2-s + (0.936 − 2.85i)3-s + (0.5 + 0.866i)4-s + (1.93 + 1.11i)5-s + (−5 + 4.47i)6-s + (3.5 + 6.06i)7-s + 6.70i·8-s + (−7.24 − 5.33i)9-s + (−2.5 − 4.33i)10-s + (9.68 − 5.59i)11-s + (2.93 − 0.614i)12-s − 2·13-s − 15.6i·14-s + (5 − 4.47i)15-s + (9.5 − 16.4i)16-s + (−23.2 + 13.4i)17-s + ⋯
L(s)  = 1  + (−0.968 − 0.559i)2-s + (0.312 − 0.950i)3-s + (0.125 + 0.216i)4-s + (0.387 + 0.223i)5-s + (−0.833 + 0.745i)6-s + (0.5 + 0.866i)7-s + 0.838i·8-s + (−0.805 − 0.593i)9-s + (−0.250 − 0.433i)10-s + (0.880 − 0.508i)11-s + (0.244 − 0.0511i)12-s − 0.153·13-s − 1.11i·14-s + (0.333 − 0.298i)15-s + (0.593 − 1.02i)16-s + (−1.36 + 0.789i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(21\)    =    \(3 \cdot 7\)
\( \varepsilon \)  =  $0.266 + 0.963i$
motivic weight  =  \(2\)
character  :  $\chi_{21} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 21,\ (\ :1),\ 0.266 + 0.963i)$
$L(\frac{3}{2})$  $\approx$  $0.511452 - 0.389330i$
$L(\frac12)$  $\approx$  $0.511452 - 0.389330i$
$L(2)$   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 \( 1 + (-0.936 + 2.85i)T \)
7 \( 1 + (-3.5 - 6.06i)T \)
good2 \( 1 + (1.93 + 1.11i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (-1.93 - 1.11i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-9.68 + 5.59i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 2T + 169T^{2} \)
17 \( 1 + (23.2 - 13.4i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (8 - 13.8i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-11.6 - 6.70i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 15.6iT - 841T^{2} \)
31 \( 1 + (-1.5 - 2.59i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (6 - 10.3i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 31.3iT - 1.68e3T^{2} \)
43 \( 1 - 44T + 1.84e3T^{2} \)
47 \( 1 + (11.6 + 6.70i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (17.4 - 10.0i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-17.4 + 10.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-13 + 22.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (26 + 45.0i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 93.9iT - 5.04e3T^{2} \)
73 \( 1 + (9 + 15.5i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 140. iT - 6.88e3T^{2} \)
89 \( 1 + (-42.6 - 24.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 93T + 9.40e3T^{2} \)
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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

−17.92195263162350222219154085379, −17.23731727751161703487350297207, −14.93470348373181760616935922438, −13.86276605426231900583664040431, −12.17389372770494872998564824719, −11.01026287403299148756144258378, −9.193521325281137063349657572869, −8.264492582949535754814509523450, −6.13694328109368701907346252175, −2.02864785952710459585102132239, 4.41658652359578847710526598533, 7.07631093751630237983571049396, 8.756563983493013863516331495487, 9.655184331960099459883477600371, 11.07786151469147311144936869268, 13.34860235969718109973705923403, 14.75428664606873376039214457949, 16.01185413912418281355287160497, 17.13641658879782553772421182721, 17.66728308610323399363629117350

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