Properties

Degree 2
Conductor $ 2 \cdot 11 $
Sign $0.648 + 0.760i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.61 − 1.17i)2-s + (−1.33 − 4.09i)3-s + (1.23 − 3.80i)4-s + (6.52 + 4.73i)5-s + (−6.97 − 5.06i)6-s + (−8.05 + 24.8i)7-s + (−2.47 − 7.60i)8-s + (6.82 − 4.95i)9-s + 16.1·10-s + (−33.3 + 14.7i)11-s − 17.2·12-s + (2.64 − 1.91i)13-s + (16.1 + 49.6i)14-s + (10.7 − 33.0i)15-s + (−12.9 − 9.40i)16-s + (16.8 + 12.2i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (−0.256 − 0.788i)3-s + (0.154 − 0.475i)4-s + (0.583 + 0.423i)5-s + (−0.474 − 0.344i)6-s + (−0.435 + 1.33i)7-s + (−0.109 − 0.336i)8-s + (0.252 − 0.183i)9-s + 0.509·10-s + (−0.914 + 0.403i)11-s − 0.414·12-s + (0.0563 − 0.0409i)13-s + (0.307 + 0.946i)14-s + (0.184 − 0.568i)15-s + (−0.202 − 0.146i)16-s + (0.240 + 0.175i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 + 0.760i)\, \overline{\Lambda}(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.648 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(22\)    =    \(2 \cdot 11\)
\( \varepsilon \)  =  $0.648 + 0.760i$
motivic weight  =  \(3\)
character  :  $\chi_{22} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 22,\ (\ :3/2),\ 0.648 + 0.760i)$
$L(2)$  $\approx$  $1.25538 - 0.579360i$
$L(\frac12)$  $\approx$  $1.25538 - 0.579360i$
$L(\frac{5}{2})$   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 \( 1 + (-1.61 + 1.17i)T \)
11 \( 1 + (33.3 - 14.7i)T \)
good3 \( 1 + (1.33 + 4.09i)T + (-21.8 + 15.8i)T^{2} \)
5 \( 1 + (-6.52 - 4.73i)T + (38.6 + 118. i)T^{2} \)
7 \( 1 + (8.05 - 24.8i)T + (-277. - 201. i)T^{2} \)
13 \( 1 + (-2.64 + 1.91i)T + (678. - 2.08e3i)T^{2} \)
17 \( 1 + (-16.8 - 12.2i)T + (1.51e3 + 4.67e3i)T^{2} \)
19 \( 1 + (38.9 + 119. i)T + (-5.54e3 + 4.03e3i)T^{2} \)
23 \( 1 - 97.8T + 1.21e4T^{2} \)
29 \( 1 + (81.5 - 250. i)T + (-1.97e4 - 1.43e4i)T^{2} \)
31 \( 1 + (161. - 117. i)T + (9.20e3 - 2.83e4i)T^{2} \)
37 \( 1 + (-112. + 347. i)T + (-4.09e4 - 2.97e4i)T^{2} \)
41 \( 1 + (-84.5 - 260. i)T + (-5.57e4 + 4.05e4i)T^{2} \)
43 \( 1 - 388.T + 7.95e4T^{2} \)
47 \( 1 + (16.0 + 49.3i)T + (-8.39e4 + 6.10e4i)T^{2} \)
53 \( 1 + (-333. + 242. i)T + (4.60e4 - 1.41e5i)T^{2} \)
59 \( 1 + (-8.12 + 24.9i)T + (-1.66e5 - 1.20e5i)T^{2} \)
61 \( 1 + (132. + 96.4i)T + (7.01e4 + 2.15e5i)T^{2} \)
67 \( 1 - 276.T + 3.00e5T^{2} \)
71 \( 1 + (-418. - 303. i)T + (1.10e5 + 3.40e5i)T^{2} \)
73 \( 1 + (74.6 - 229. i)T + (-3.14e5 - 2.28e5i)T^{2} \)
79 \( 1 + (-220. + 160. i)T + (1.52e5 - 4.68e5i)T^{2} \)
83 \( 1 + (58.7 + 42.6i)T + (1.76e5 + 5.43e5i)T^{2} \)
89 \( 1 + 1.19e3T + 7.04e5T^{2} \)
97 \( 1 + (1.18e3 - 860. i)T + (2.82e5 - 8.68e5i)T^{2} \)
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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

−17.91478948919482668202204300445, −15.81154759608634443745844081927, −14.71039489005093166359882320151, −13.01667032689774305635431671870, −12.53293475086121485769664569538, −10.91738940241206495474703035528, −9.280129257696872031992573259402, −6.90775298407155775345447574611, −5.52804045030340407986287728335, −2.46453744128779366832478711766, 4.06093736295486618416107366091, 5.65424161763885758285245969751, 7.61592026690383926944713276178, 9.771042301484117589623539995578, 10.85807627699460035804845690535, 12.95196516929332497641643053131, 13.73531942292230278791621542805, 15.32525034441322959914110059819, 16.60211221328392733758517734881, 16.96342053102986890558572169470

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