Properties

Degree 2
Conductor 11
Sign $0.621 + 0.783i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.38 − 1.09i)2-s + (12.7 − 9.28i)3-s + (−2.71 − 1.96i)4-s + (5.48 + 16.8i)5-s + (−53.4 + 17.3i)6-s + (−19.6 + 27.0i)7-s + (40.4 + 55.6i)8-s + (52.1 − 160. i)9-s − 63.1i·10-s + (44.7 + 112. i)11-s − 52.9·12-s + (−133. − 43.4i)13-s + (96.1 − 69.8i)14-s + (226. + 164. i)15-s + (−59.0 − 181. i)16-s + (84.2 − 27.3i)17-s + ⋯
L(s)  = 1  + (−0.845 − 0.274i)2-s + (1.42 − 1.03i)3-s + (−0.169 − 0.123i)4-s + (0.219 + 0.675i)5-s + (−1.48 + 0.482i)6-s + (−0.400 + 0.551i)7-s + (0.632 + 0.869i)8-s + (0.643 − 1.98i)9-s − 0.631i·10-s + (0.369 + 0.929i)11-s − 0.367·12-s + (−0.790 − 0.256i)13-s + (0.490 − 0.356i)14-s + (1.00 + 0.732i)15-s + (−0.230 − 0.710i)16-s + (0.291 − 0.0946i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(5-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(11\)
\( \varepsilon \)  =  $0.621 + 0.783i$
motivic weight  =  \(4\)
character  :  $\chi_{11} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 11,\ (\ :2),\ 0.621 + 0.783i)$
$L(\frac{5}{2})$  $\approx$  $0.871480 - 0.420893i$
$L(\frac12)$  $\approx$  $0.871480 - 0.420893i$
$L(3)$   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 \( 1 + (-44.7 - 112. i)T \)
good2 \( 1 + (3.38 + 1.09i)T + (12.9 + 9.40i)T^{2} \)
3 \( 1 + (-12.7 + 9.28i)T + (25.0 - 77.0i)T^{2} \)
5 \( 1 + (-5.48 - 16.8i)T + (-505. + 367. i)T^{2} \)
7 \( 1 + (19.6 - 27.0i)T + (-741. - 2.28e3i)T^{2} \)
13 \( 1 + (133. + 43.4i)T + (2.31e4 + 1.67e4i)T^{2} \)
17 \( 1 + (-84.2 + 27.3i)T + (6.75e4 - 4.90e4i)T^{2} \)
19 \( 1 + (192. + 264. i)T + (-4.02e4 + 1.23e5i)T^{2} \)
23 \( 1 + 527.T + 2.79e5T^{2} \)
29 \( 1 + (100. - 138. i)T + (-2.18e5 - 6.72e5i)T^{2} \)
31 \( 1 + (318. - 981. i)T + (-7.47e5 - 5.42e5i)T^{2} \)
37 \( 1 + (-238. - 173. i)T + (5.79e5 + 1.78e6i)T^{2} \)
41 \( 1 + (-16.1 - 22.2i)T + (-8.73e5 + 2.68e6i)T^{2} \)
43 \( 1 + 2.50e3iT - 3.41e6T^{2} \)
47 \( 1 + (-1.08e3 + 785. i)T + (1.50e6 - 4.64e6i)T^{2} \)
53 \( 1 + (-159. + 491. i)T + (-6.38e6 - 4.63e6i)T^{2} \)
59 \( 1 + (-1.79e3 - 1.30e3i)T + (3.74e6 + 1.15e7i)T^{2} \)
61 \( 1 + (-5.06e3 + 1.64e3i)T + (1.12e7 - 8.13e6i)T^{2} \)
67 \( 1 + 4.02e3T + 2.01e7T^{2} \)
71 \( 1 + (-538. - 1.65e3i)T + (-2.05e7 + 1.49e7i)T^{2} \)
73 \( 1 + (-1.32e3 + 1.82e3i)T + (-8.77e6 - 2.70e7i)T^{2} \)
79 \( 1 + (-3.33e3 - 1.08e3i)T + (3.15e7 + 2.28e7i)T^{2} \)
83 \( 1 + (1.05e4 - 3.42e3i)T + (3.83e7 - 2.78e7i)T^{2} \)
89 \( 1 - 2.77e3T + 6.27e7T^{2} \)
97 \( 1 + (-3.04e3 + 9.38e3i)T + (-7.16e7 - 5.20e7i)T^{2} \)
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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

−19.46813493408203915302477622517, −18.53973923380379159456876510718, −17.65811878235009786123398911002, −14.95919617758492132898476870776, −14.03047383717371924021647076150, −12.46589188566179489025569585308, −9.953131988191552200881844098853, −8.735236192433585620935819667396, −7.15680928599117016741818066737, −2.30452235969826111214534811765, 3.99843181538594636360595410056, 7.984619358551889532129356863252, 9.144257452871662531740908820699, 10.09298484565305749821942742712, 13.19933792033549000156349917319, 14.44729758261952974629097407181, 16.21478909939041780300710033268, 16.87105644316337875510711536154, 18.99344528998970562662406273913, 19.90682094385855768562497417430

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