Properties

Degree 2
Conductor 11
Sign $-0.971 + 0.236i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.45 + 1.77i)2-s + (−10.4 − 7.55i)3-s + (13.6 − 9.95i)4-s + (−9.33 + 28.7i)5-s + (70.1 + 22.8i)6-s + (−33.6 − 46.2i)7-s + (−3.13 + 4.31i)8-s + (26.0 + 80.2i)9-s − 173. i·10-s + (−104. + 61.2i)11-s − 217.·12-s + (−21.1 + 6.88i)13-s + (265. + 192. i)14-s + (314. − 228. i)15-s + (−74.2 + 228. i)16-s + (−176. − 57.3i)17-s + ⋯
L(s)  = 1  + (−1.36 + 0.443i)2-s + (−1.15 − 0.839i)3-s + (0.856 − 0.621i)4-s + (−0.373 + 1.14i)5-s + (1.94 + 0.633i)6-s + (−0.685 − 0.944i)7-s + (−0.0490 + 0.0674i)8-s + (0.321 + 0.990i)9-s − 1.73i·10-s + (−0.862 + 0.506i)11-s − 1.51·12-s + (−0.125 + 0.0407i)13-s + (1.35 + 0.984i)14-s + (1.39 − 1.01i)15-s + (−0.290 + 0.892i)16-s + (−0.610 − 0.198i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.236i)\, \overline{\Lambda}(5-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(11\)
\( \varepsilon \)  =  $-0.971 + 0.236i$
motivic weight  =  \(4\)
character  :  $\chi_{11} (6, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 11,\ (\ :2),\ -0.971 + 0.236i)$
$L(\frac{5}{2})$  $\approx$  $0.00292430 - 0.0243713i$
$L(\frac12)$  $\approx$  $0.00292430 - 0.0243713i$
$L(3)$   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 \( 1 + (104. - 61.2i)T \)
good2 \( 1 + (5.45 - 1.77i)T + (12.9 - 9.40i)T^{2} \)
3 \( 1 + (10.4 + 7.55i)T + (25.0 + 77.0i)T^{2} \)
5 \( 1 + (9.33 - 28.7i)T + (-505. - 367. i)T^{2} \)
7 \( 1 + (33.6 + 46.2i)T + (-741. + 2.28e3i)T^{2} \)
13 \( 1 + (21.1 - 6.88i)T + (2.31e4 - 1.67e4i)T^{2} \)
17 \( 1 + (176. + 57.3i)T + (6.75e4 + 4.90e4i)T^{2} \)
19 \( 1 + (-169. + 233. i)T + (-4.02e4 - 1.23e5i)T^{2} \)
23 \( 1 + 116.T + 2.79e5T^{2} \)
29 \( 1 + (419. + 576. i)T + (-2.18e5 + 6.72e5i)T^{2} \)
31 \( 1 + (28.3 + 87.2i)T + (-7.47e5 + 5.42e5i)T^{2} \)
37 \( 1 + (1.02e3 - 744. i)T + (5.79e5 - 1.78e6i)T^{2} \)
41 \( 1 + (-340. + 468. i)T + (-8.73e5 - 2.68e6i)T^{2} \)
43 \( 1 - 1.41e3iT - 3.41e6T^{2} \)
47 \( 1 + (3.36e3 + 2.44e3i)T + (1.50e6 + 4.64e6i)T^{2} \)
53 \( 1 + (-1.13e3 - 3.49e3i)T + (-6.38e6 + 4.63e6i)T^{2} \)
59 \( 1 + (378. - 275. i)T + (3.74e6 - 1.15e7i)T^{2} \)
61 \( 1 + (-777. - 252. i)T + (1.12e7 + 8.13e6i)T^{2} \)
67 \( 1 - 6.01e3T + 2.01e7T^{2} \)
71 \( 1 + (-1.14e3 + 3.53e3i)T + (-2.05e7 - 1.49e7i)T^{2} \)
73 \( 1 + (1.26e3 + 1.74e3i)T + (-8.77e6 + 2.70e7i)T^{2} \)
79 \( 1 + (1.03e4 - 3.37e3i)T + (3.15e7 - 2.28e7i)T^{2} \)
83 \( 1 + (4.23e3 + 1.37e3i)T + (3.83e7 + 2.78e7i)T^{2} \)
89 \( 1 + 1.58e3T + 6.27e7T^{2} \)
97 \( 1 + (5.78e3 + 1.78e4i)T + (-7.16e7 + 5.20e7i)T^{2} \)
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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

−18.66179269835816651759483036973, −17.93794226106205200974526650346, −16.91306313657320989283056141287, −15.58568154922842410367284578988, −13.20186017783644510443667800608, −11.22405681304792590699486095849, −10.11102679916742680214758608409, −7.42966270034087887983702054793, −6.71362726836937285063502453491, −0.04768232989740300483805046856, 5.33375273399099302973235601308, 8.527591908548574722969546036499, 9.825992453622954164315950861234, 11.21547669498278198826400858278, 12.50788758811442428629595296671, 15.93007131023420052189089158790, 16.37662744716281736341164716226, 17.68274897766651841029885652842, 18.95957128874381145428663968135, 20.33948896866492285687845583756

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