Properties

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

Origins

Related objects

Downloads

Learn more about

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  :  $\chi_{11} (2, \cdot )$
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 \( 1 + (-4.66e3 + 1.43e4i)T + (-7.16e7 - 5.20e7i)T^{2} \)
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

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

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