Properties

Degree 2
Conductor 11
Sign $0.0909 - 0.995i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 5.47i·2-s − 3·3-s − 14·4-s + 31·5-s − 16.4i·6-s − 54.7i·7-s + 10.9i·8-s − 72·9-s + 169. i·10-s + (11 − 120. i)11-s + 42·12-s + 186. i·13-s + 300·14-s − 93·15-s − 284·16-s − 230. i·17-s + ⋯
L(s)  = 1  + 1.36i·2-s − 0.333·3-s − 0.875·4-s + 1.23·5-s − 0.456i·6-s − 1.11i·7-s + 0.171i·8-s − 0.888·9-s + 1.69i·10-s + (0.0909 − 0.995i)11-s + 0.291·12-s + 1.10i·13-s + 1.53·14-s − 0.413·15-s − 1.10·16-s − 0.795i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0909 - 0.995i)\, \overline{\Lambda}(5-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0909 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(11\)
\( \varepsilon \)  =  $0.0909 - 0.995i$
motivic weight  =  \(4\)
character  :  $\chi_{11} (10, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 11,\ (\ :2),\ 0.0909 - 0.995i)$
$L(\frac{5}{2})$  $\approx$  $0.794643 + 0.725407i$
$L(\frac12)$  $\approx$  $0.794643 + 0.725407i$
$L(3)$   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 \( 1 + (-11 + 120. i)T \)
good2 \( 1 - 5.47iT - 16T^{2} \)
3 \( 1 + 3T + 81T^{2} \)
5 \( 1 - 31T + 625T^{2} \)
7 \( 1 + 54.7iT - 2.40e3T^{2} \)
13 \( 1 - 186. iT - 2.85e4T^{2} \)
17 \( 1 + 230. iT - 8.35e4T^{2} \)
19 \( 1 - 98.5iT - 1.30e5T^{2} \)
23 \( 1 - 277T + 2.79e5T^{2} \)
29 \( 1 - 1.27e3iT - 7.07e5T^{2} \)
31 \( 1 + 1.36e3T + 9.23e5T^{2} \)
37 \( 1 - 167T + 1.87e6T^{2} \)
41 \( 1 - 1.06e3iT - 2.82e6T^{2} \)
43 \( 1 + 1.20e3iT - 3.41e6T^{2} \)
47 \( 1 - 1.70e3T + 4.87e6T^{2} \)
53 \( 1 - 4.52e3T + 7.89e6T^{2} \)
59 \( 1 + 2.36e3T + 1.21e7T^{2} \)
61 \( 1 + 3.96e3iT - 1.38e7T^{2} \)
67 \( 1 + 2.80e3T + 2.01e7T^{2} \)
71 \( 1 - 3.39e3T + 2.54e7T^{2} \)
73 \( 1 - 3.31e3iT - 2.83e7T^{2} \)
79 \( 1 - 6.09e3iT - 3.89e7T^{2} \)
83 \( 1 + 832. iT - 4.74e7T^{2} \)
89 \( 1 + 4.67e3T + 6.27e7T^{2} \)
97 \( 1 - 4.24e3T + 8.85e7T^{2} \)
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

−20.26132514032608998620177529834, −18.18253336283652801360653841472, −16.87079131996468627874928504157, −16.56304172213857674847604833284, −14.28695245088176938995164735449, −13.74945259582314837992793239083, −11.05392227188633231096315632129, −8.991403689752017235867616356036, −6.90260280784055901568593389407, −5.53102188592883621196977418424, 2.36292077307959260973677058117, 5.73105080156715589507556740723, 9.204238464977936325510442516002, 10.52479981427678601601763710300, 12.05413188563647012617656866518, 13.16983060155429795253515143690, 15.07365855119871938737353357775, 17.34444638482209044977961468755, 18.23883335823624907732354596626, 19.76891243336154225506294972049

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