Properties

Degree 2
Conductor 11
Sign $0.964 + 0.262i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.67 − 1.19i)2-s + (−1.64 − 1.19i)3-s + (−0.850 + 0.618i)4-s + (−1.76 + 5.43i)5-s + (−7.47 − 2.43i)6-s + (−2.52 − 3.47i)7-s + (−38.7 + 53.3i)8-s + (−23.7 − 73.1i)9-s + 22.1i·10-s + (120. − 3.19i)11-s + 2.13·12-s + (189. − 61.5i)13-s + (−13.4 − 9.77i)14-s + (9.41 − 6.83i)15-s + (−73.5 + 226. i)16-s + (−269. − 87.5i)17-s + ⋯
L(s)  = 1  + (0.919 − 0.298i)2-s + (−0.182 − 0.132i)3-s + (−0.0531 + 0.0386i)4-s + (−0.0706 + 0.217i)5-s + (−0.207 − 0.0675i)6-s + (−0.0515 − 0.0710i)7-s + (−0.605 + 0.833i)8-s + (−0.293 − 0.902i)9-s + 0.221i·10-s + (0.999 − 0.0264i)11-s + 0.0148·12-s + (1.12 − 0.364i)13-s + (−0.0686 − 0.0498i)14-s + (0.0418 − 0.0303i)15-s + (−0.287 + 0.884i)16-s + (−0.932 − 0.302i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.262i)\, \overline{\Lambda}(5-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.964 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(11\)
\( \varepsilon \)  =  $0.964 + 0.262i$
motivic weight  =  \(4\)
character  :  $\chi_{11} (6, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 11,\ (\ :2),\ 0.964 + 0.262i)$
$L(\frac{5}{2})$  $\approx$  $1.37220 - 0.183408i$
$L(\frac12)$  $\approx$  $1.37220 - 0.183408i$
$L(3)$   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 \( 1 + (-120. + 3.19i)T \)
good2 \( 1 + (-3.67 + 1.19i)T + (12.9 - 9.40i)T^{2} \)
3 \( 1 + (1.64 + 1.19i)T + (25.0 + 77.0i)T^{2} \)
5 \( 1 + (1.76 - 5.43i)T + (-505. - 367. i)T^{2} \)
7 \( 1 + (2.52 + 3.47i)T + (-741. + 2.28e3i)T^{2} \)
13 \( 1 + (-189. + 61.5i)T + (2.31e4 - 1.67e4i)T^{2} \)
17 \( 1 + (269. + 87.5i)T + (6.75e4 + 4.90e4i)T^{2} \)
19 \( 1 + (237. - 326. i)T + (-4.02e4 - 1.23e5i)T^{2} \)
23 \( 1 + 189.T + 2.79e5T^{2} \)
29 \( 1 + (797. + 1.09e3i)T + (-2.18e5 + 6.72e5i)T^{2} \)
31 \( 1 + (-421. - 1.29e3i)T + (-7.47e5 + 5.42e5i)T^{2} \)
37 \( 1 + (-690. + 501. i)T + (5.79e5 - 1.78e6i)T^{2} \)
41 \( 1 + (-218. + 301. i)T + (-8.73e5 - 2.68e6i)T^{2} \)
43 \( 1 + 854. iT - 3.41e6T^{2} \)
47 \( 1 + (-1.28e3 - 932. i)T + (1.50e6 + 4.64e6i)T^{2} \)
53 \( 1 + (274. + 846. i)T + (-6.38e6 + 4.63e6i)T^{2} \)
59 \( 1 + (4.67e3 - 3.39e3i)T + (3.74e6 - 1.15e7i)T^{2} \)
61 \( 1 + (3.71e3 + 1.20e3i)T + (1.12e7 + 8.13e6i)T^{2} \)
67 \( 1 - 6.19e3T + 2.01e7T^{2} \)
71 \( 1 + (-1.65e3 + 5.08e3i)T + (-2.05e7 - 1.49e7i)T^{2} \)
73 \( 1 + (-3.57e3 - 4.91e3i)T + (-8.77e6 + 2.70e7i)T^{2} \)
79 \( 1 + (4.60e3 - 1.49e3i)T + (3.15e7 - 2.28e7i)T^{2} \)
83 \( 1 + (2.42e3 + 786. i)T + (3.83e7 + 2.78e7i)T^{2} \)
89 \( 1 + 1.60e3T + 6.27e7T^{2} \)
97 \( 1 + (-4.66e3 - 1.43e4i)T + (-7.16e7 + 5.20e7i)T^{2} \)
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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

−20.18340180237229313286837065423, −18.35751646916624459131778517270, −17.15541236312107533359220481780, −15.16217101728818632500448926788, −13.91361544297612450832502650256, −12.51439389200200025755316039602, −11.27124156641500563597618338992, −8.849061861840539049768757603656, −6.18891829672656497768477447539, −3.78568739047174893620691187189, 4.41089944555135701446353188788, 6.28960553433772509791866433840, 8.956133093165205190387217796286, 11.14283464342644179549610454479, 12.96235513798851996397888653575, 14.09377431163961807109247397831, 15.51033351696155503037667107951, 16.84954508482684422356878683171, 18.59706267389342360291109629441, 20.02039078058329356130037488669

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