Properties

Degree 2
Conductor $ 3 \cdot 7 $
Sign $0.928 - 0.371i$
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 + (−2.93 − 0.614i)3-s + (0.5 + 0.866i)4-s + (−1.93 − 1.11i)5-s + (−5.00 − 4.47i)6-s + (3.5 + 6.06i)7-s − 6.70i·8-s + (8.24 + 3.60i)9-s + (−2.5 − 4.33i)10-s + (−9.68 + 5.59i)11-s + (−0.936 − 2.85i)12-s − 2·13-s + 15.6i·14-s + (5.00 + 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 + (−0.978 − 0.204i)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 + (0.916 + 0.400i)9-s + (−0.250 − 0.433i)10-s + (−0.880 + 0.508i)11-s + (−0.0780 − 0.237i)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

\[\begin{aligned} \Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(21\)    =    \(3 \cdot 7\)
\( \varepsilon \)  =  $0.928 - 0.371i$
motivic weight  =  \(2\)
character  :  $\chi_{21} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 21,\ (\ :1),\ 0.928 - 0.371i)$
$L(\frac{3}{2})$  $\approx$  $0.992942 + 0.191397i$
$L(\frac12)$  $\approx$  $0.992942 + 0.191397i$
$L(2)$   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 \( 1 + (2.93 + 0.614i)T \)
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

−18.08321292704430157315563847028, −16.41054882307387875101483910467, −15.56618379942246307775577258454, −14.32389765542925139278161845480, −12.69062168776067391165718117538, −11.96745805139525695531750575152, −10.11171694228189508306715071413, −7.66361245564424492733683551836, −5.86317044138694108514974959027, −4.77249049080509833323078401705, 3.94542690833329969833691836104, 5.48903860934316606884924599157, 7.73141960301047997538788715575, 10.47036081354431830917804358796, 11.36073541210854960998510300557, 12.56380924719453118065801914145, 13.78091377295046842559865238585, 15.22146390234541627482057730647, 16.75070961773870300137640036501, 17.70849544448348399096742239171

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