↑

Properties

Label 2-22-11.10-c14-0-2
Degree $2$
Conductor $22$
Sign $-0.654 - 0.756i$
Analytic cond. $27.3523$
Root an. cond. $5.22994$
Motivic weight $14$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 90.5i·2-s − 3.24e3·3-s − 8.19e3·4-s − 6.68e3·5-s − 2.94e5i·6-s − 2.86e5i·7-s − 7.41e5i·8-s + 5.77e6·9-s − 6.04e5i·10-s + (1.47e7 − 1.27e7i)11-s + 2.66e7·12-s − 5.04e7i·13-s + 2.58e7·14-s + 2.17e7·15-s + 6.71e7·16-s + 1.90e8i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.48·3-s − 0.500·4-s − 0.0855·5-s − 1.05i·6-s − 0.347i·7-s − 0.353i·8-s + 1.20·9-s − 0.0604i·10-s + (0.756 − 0.654i)11-s + 0.742·12-s − 0.803i·13-s + 0.245·14-s + 0.127·15-s + 0.250·16-s + 0.464i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.756i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.654 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $-0.654 - 0.756i$
Analytic conductor: \(27.3523\)
Root analytic conductor: \(5.22994\)
Motivic weight: \(14\)
Rational: no
Arithmetic: yes
Character: $\chi_{22} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 22,\ (\ :7),\ -0.654 - 0.756i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.5357736194\)
\(L(\frac12)\) \(\approx\) \(0.5357736194\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 90.5iT \)
11 \( 1 + (-1.47e7 + 1.27e7i)T \)
good3 \( 1 + 3.24e3T + 4.78e6T^{2} \)
5 \( 1 + 6.68e3T + 6.10e9T^{2} \)
7 \( 1 + 2.86e5iT - 6.78e11T^{2} \)
13 \( 1 + 5.04e7iT - 3.93e15T^{2} \)
17 \( 1 - 1.90e8iT - 1.68e17T^{2} \)
19 \( 1 - 4.06e8iT - 7.99e17T^{2} \)
23 \( 1 + 2.91e9T + 1.15e19T^{2} \)
29 \( 1 + 1.38e10iT - 2.97e20T^{2} \)
31 \( 1 - 4.01e7T + 7.56e20T^{2} \)
37 \( 1 + 7.99e10T + 9.01e21T^{2} \)
41 \( 1 - 2.73e11iT - 3.79e22T^{2} \)
43 \( 1 - 3.24e11iT - 7.38e22T^{2} \)
47 \( 1 - 3.95e11T + 2.56e23T^{2} \)
53 \( 1 - 2.76e10T + 1.37e24T^{2} \)
59 \( 1 + 8.11e11T + 6.19e24T^{2} \)
61 \( 1 + 1.19e11iT - 9.87e24T^{2} \)
67 \( 1 - 9.97e11T + 3.67e25T^{2} \)
71 \( 1 - 1.53e13T + 8.27e25T^{2} \)
73 \( 1 - 1.29e12iT - 1.22e26T^{2} \)
79 \( 1 - 2.71e13iT - 3.68e26T^{2} \)
83 \( 1 - 4.12e13iT - 7.36e26T^{2} \)
89 \( 1 + 4.41e13T + 1.95e27T^{2} \)
97 \( 1 + 5.84e13T + 6.52e27T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.46832215194130323370635055058, −13.90971509735878503000413684256, −12.44329241934834392643545733341, −11.25999929129428063734098494366, −9.975954618743547540842734668230, −8.030503482401609996408803212141, −6.43673223361131739636647092183, −5.59008329798081421398370804830, −4.02189762242141521215173236017, −0.952311557750251638138716450303, 0.28148925222632453705540448363, 1.84283299679148058346412909954, 4.13764940017531533681023732349, 5.46858194508370383071678985184, 6.92001142181290897511034950610, 9.173278947605516914136714975722, 10.53423795713471213415116921753, 11.78056408434607397765315330420, 12.23492431474223116207031016119, 13.97036295193085079619318244311

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