Properties

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

Origins

Related objects

Downloads

Learn more about

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  :  $\chi_{22} (15, \cdot )$
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 \( 1 + (1.18e3 + 860. i)T + (2.82e5 + 8.68e5i)T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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