Properties

Degree 2
Conductor $ 2 \cdot 11 $
Sign $-0.629 - 0.777i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.618 + 1.90i)2-s + (−6.12 + 4.45i)3-s + (−3.23 − 2.35i)4-s + (1.67 + 5.14i)5-s + (−4.68 − 14.4i)6-s + (17.9 + 13.0i)7-s + (6.47 − 4.70i)8-s + (9.39 − 28.9i)9-s − 10.8·10-s + (−11.0 + 34.7i)11-s + 30.3·12-s + (23.7 − 73.0i)13-s + (−35.8 + 26.0i)14-s + (−33.1 − 24.0i)15-s + (4.94 + 15.2i)16-s + (18.3 + 56.3i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−1.17 + 0.856i)3-s + (−0.404 − 0.293i)4-s + (0.149 + 0.460i)5-s + (−0.318 − 0.980i)6-s + (0.967 + 0.702i)7-s + (0.286 − 0.207i)8-s + (0.347 − 1.07i)9-s − 0.342·10-s + (−0.302 + 0.953i)11-s + 0.728·12-s + (0.506 − 1.55i)13-s + (−0.684 + 0.497i)14-s + (−0.570 − 0.414i)15-s + (0.0772 + 0.237i)16-s + (0.261 + 0.804i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 - 0.777i)\, \overline{\Lambda}(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(22\)    =    \(2 \cdot 11\)
\( \varepsilon \)  =  $-0.629 - 0.777i$
motivic weight  =  \(3\)
character  :  $\chi_{22} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 22,\ (\ :3/2),\ -0.629 - 0.777i)$
$L(2)$  $\approx$  $0.313334 + 0.656778i$
$L(\frac12)$  $\approx$  $0.313334 + 0.656778i$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;11\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + (0.618 - 1.90i)T \)
11 \( 1 + (11.0 - 34.7i)T \)
good3 \( 1 + (6.12 - 4.45i)T + (8.34 - 25.6i)T^{2} \)
5 \( 1 + (-1.67 - 5.14i)T + (-101. + 73.4i)T^{2} \)
7 \( 1 + (-17.9 - 13.0i)T + (105. + 326. i)T^{2} \)
13 \( 1 + (-23.7 + 73.0i)T + (-1.77e3 - 1.29e3i)T^{2} \)
17 \( 1 + (-18.3 - 56.3i)T + (-3.97e3 + 2.88e3i)T^{2} \)
19 \( 1 + (77.0 - 55.9i)T + (2.11e3 - 6.52e3i)T^{2} \)
23 \( 1 - 142.T + 1.21e4T^{2} \)
29 \( 1 + (-16.5 - 12.0i)T + (7.53e3 + 2.31e4i)T^{2} \)
31 \( 1 + (-65.9 + 202. i)T + (-2.41e4 - 1.75e4i)T^{2} \)
37 \( 1 + (-117. - 85.5i)T + (1.56e4 + 4.81e4i)T^{2} \)
41 \( 1 + (67.0 - 48.6i)T + (2.12e4 - 6.55e4i)T^{2} \)
43 \( 1 + 151.T + 7.95e4T^{2} \)
47 \( 1 + (73.1 - 53.1i)T + (3.20e4 - 9.87e4i)T^{2} \)
53 \( 1 + (72.5 - 223. i)T + (-1.20e5 - 8.75e4i)T^{2} \)
59 \( 1 + (244. + 177. i)T + (6.34e4 + 1.95e5i)T^{2} \)
61 \( 1 + (46.1 + 141. i)T + (-1.83e5 + 1.33e5i)T^{2} \)
67 \( 1 - 826.T + 3.00e5T^{2} \)
71 \( 1 + (277. + 854. i)T + (-2.89e5 + 2.10e5i)T^{2} \)
73 \( 1 + (-111. - 81.1i)T + (1.20e5 + 3.69e5i)T^{2} \)
79 \( 1 + (94.0 - 289. i)T + (-3.98e5 - 2.89e5i)T^{2} \)
83 \( 1 + (236. + 726. i)T + (-4.62e5 + 3.36e5i)T^{2} \)
89 \( 1 + 313.T + 7.04e5T^{2} \)
97 \( 1 + (180. - 553. i)T + (-7.38e5 - 5.36e5i)T^{2} \)
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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

−17.67863521607243215670994087111, −16.89157980752764458458304136574, −15.30330045779383103834161107708, −14.95027215575722735522625753928, −12.66446691969587191836063966688, −11.01105744352734982325271591965, −10.12867474646955865261162915617, −8.160973284074851906535174974462, −6.06402805807492753657783912086, −4.87669696456338650979064339367, 1.11978324879953669972392833125, 4.91856253585733280620776097713, 6.91257774629780427196959960341, 8.749319767119791034890050419680, 10.96982672320137755576334432116, 11.50674989118800180014276782730, 12.95013434486640666311543577358, 14.00638485873588345696455320639, 16.52517060731106987808948979171, 17.20317249667413896873862529866

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