Properties

Degree 2
Conductor 11
Sign $0.673 - 0.739i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.10 + 5.65i)2-s + (−4.15 − 12.7i)3-s + (−10.1 + 31.2i)4-s + (−10.0 − 7.30i)5-s + (55.2 − 75.9i)6-s + (23.8 + 7.73i)7-s + (−111. + 36.2i)8-s + (−80.7 + 58.6i)9-s − 86.8i·10-s + (16.1 + 119. i)11-s + 441.·12-s + (−110. − 151. i)13-s + (54.0 + 166. i)14-s + (−51.6 + 158. i)15-s + (−239. − 173. i)16-s + (−30.5 + 42.0i)17-s + ⋯
L(s)  = 1  + (1.02 + 1.41i)2-s + (−0.461 − 1.42i)3-s + (−0.633 + 1.95i)4-s + (−0.402 − 0.292i)5-s + (1.53 − 2.11i)6-s + (0.485 + 0.157i)7-s + (−1.74 + 0.566i)8-s + (−0.996 + 0.724i)9-s − 0.868i·10-s + (0.133 + 0.991i)11-s + 3.06·12-s + (−0.653 − 0.898i)13-s + (0.275 + 0.848i)14-s + (−0.229 + 0.706i)15-s + (−0.933 − 0.678i)16-s + (−0.105 + 0.145i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(5-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(11\)
\( \varepsilon \)  =  $0.673 - 0.739i$
motivic weight  =  \(4\)
character  :  $\chi_{11} (8, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 11,\ (\ :2),\ 0.673 - 0.739i)$
$L(\frac{5}{2})$  $\approx$  $1.27894 + 0.565046i$
$L(\frac12)$  $\approx$  $1.27894 + 0.565046i$
$L(3)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 11$, \(F_p\) is a polynomial of degree 2. If $p = 11$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad11 \( 1 + (-16.1 - 119. i)T \)
good2 \( 1 + (-4.10 - 5.65i)T + (-4.94 + 15.2i)T^{2} \)
3 \( 1 + (4.15 + 12.7i)T + (-65.5 + 47.6i)T^{2} \)
5 \( 1 + (10.0 + 7.30i)T + (193. + 594. i)T^{2} \)
7 \( 1 + (-23.8 - 7.73i)T + (1.94e3 + 1.41e3i)T^{2} \)
13 \( 1 + (110. + 151. i)T + (-8.82e3 + 2.71e4i)T^{2} \)
17 \( 1 + (30.5 - 42.0i)T + (-2.58e4 - 7.94e4i)T^{2} \)
19 \( 1 + (-369. + 119. i)T + (1.05e5 - 7.66e4i)T^{2} \)
23 \( 1 - 164.T + 2.79e5T^{2} \)
29 \( 1 + (-596. - 193. i)T + (5.72e5 + 4.15e5i)T^{2} \)
31 \( 1 + (974. - 708. i)T + (2.85e5 - 8.78e5i)T^{2} \)
37 \( 1 + (-203. + 627. i)T + (-1.51e6 - 1.10e6i)T^{2} \)
41 \( 1 + (2.18e3 - 710. i)T + (2.28e6 - 1.66e6i)T^{2} \)
43 \( 1 + 1.74e3iT - 3.41e6T^{2} \)
47 \( 1 + (-1.14e3 - 3.51e3i)T + (-3.94e6 + 2.86e6i)T^{2} \)
53 \( 1 + (-63.9 + 46.4i)T + (2.43e6 - 7.50e6i)T^{2} \)
59 \( 1 + (-341. + 1.04e3i)T + (-9.80e6 - 7.12e6i)T^{2} \)
61 \( 1 + (-137. + 188. i)T + (-4.27e6 - 1.31e7i)T^{2} \)
67 \( 1 - 6.39e3T + 2.01e7T^{2} \)
71 \( 1 + (-4.46e3 - 3.24e3i)T + (7.85e6 + 2.41e7i)T^{2} \)
73 \( 1 + (-2.97e3 - 968. i)T + (2.29e7 + 1.66e7i)T^{2} \)
79 \( 1 + (1.85e3 + 2.55e3i)T + (-1.20e7 + 3.70e7i)T^{2} \)
83 \( 1 + (-3.66e3 + 5.04e3i)T + (-1.46e7 - 4.51e7i)T^{2} \)
89 \( 1 + 1.26e3T + 6.27e7T^{2} \)
97 \( 1 + (-4.42e3 + 3.21e3i)T + (2.73e7 - 8.41e7i)T^{2} \)
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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

−20.00084134542239903662773318395, −18.03231395584702994876112801425, −17.27523933486539120386879007749, −15.65719420605356773182733067469, −14.31136353530716164834569655821, −12.87139827063226194860338553316, −12.08584164358793033694688922435, −7.957421545799656679234928090308, −6.94001776307360107486864055567, −5.16504122985357109131641935826, 3.68376350884209958541191495627, 5.13897436698524551184570730890, 9.664084914704939508433114867401, 11.03834934050303985910598188399, 11.72356241553181136911422907487, 13.88224356496153776880702889869, 15.06946261333811521491368105428, 16.63983460892448176680251995730, 18.86096175099561081495372488719, 20.21604572458361656859965318798

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