Properties

Label 2-22-11.7-c8-0-5
Degree $2$
Conductor $22$
Sign $0.0103 + 0.999i$
Analytic cond. $8.96232$
Root an. cond. $2.99371$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.65 + 9.15i)2-s + (−24.3 + 74.9i)3-s + (−39.5 − 121. i)4-s + (−467. + 339. i)5-s + (−524. − 721. i)6-s + (−1.19e3 + 388. i)7-s + (1.37e3 + 447. i)8-s + (278. + 202. i)9-s − 6.53e3i·10-s + (−1.16e3 − 1.45e4i)11-s + 1.00e4·12-s + (1.30e4 − 1.79e4i)13-s + (4.39e3 − 1.35e4i)14-s + (−1.40e4 − 4.33e4i)15-s + (−1.32e4 + 9.63e3i)16-s + (−5.19e4 − 7.15e4i)17-s + ⋯
L(s)  = 1  + (−0.415 + 0.572i)2-s + (−0.300 + 0.925i)3-s + (−0.154 − 0.475i)4-s + (−0.747 + 0.543i)5-s + (−0.404 − 0.556i)6-s + (−0.498 + 0.161i)7-s + (0.336 + 0.109i)8-s + (0.0425 + 0.0308i)9-s − 0.653i·10-s + (−0.0797 − 0.996i)11-s + 0.486·12-s + (0.457 − 0.629i)13-s + (0.114 − 0.352i)14-s + (−0.278 − 0.855i)15-s + (−0.202 + 0.146i)16-s + (−0.622 − 0.856i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0103 + 0.999i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.0103 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $0.0103 + 0.999i$
Analytic conductor: \(8.96232\)
Root analytic conductor: \(2.99371\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{22} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 22,\ (\ :4),\ 0.0103 + 0.999i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.00783658 - 0.00775625i\)
\(L(\frac12)\) \(\approx\) \(0.00783658 - 0.00775625i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (6.65 - 9.15i)T \)
11 \( 1 + (1.16e3 + 1.45e4i)T \)
good3 \( 1 + (24.3 - 74.9i)T + (-5.30e3 - 3.85e3i)T^{2} \)
5 \( 1 + (467. - 339. i)T + (1.20e5 - 3.71e5i)T^{2} \)
7 \( 1 + (1.19e3 - 388. i)T + (4.66e6 - 3.38e6i)T^{2} \)
13 \( 1 + (-1.30e4 + 1.79e4i)T + (-2.52e8 - 7.75e8i)T^{2} \)
17 \( 1 + (5.19e4 + 7.15e4i)T + (-2.15e9 + 6.63e9i)T^{2} \)
19 \( 1 + (-2.21e4 - 7.20e3i)T + (1.37e10 + 9.98e9i)T^{2} \)
23 \( 1 + 2.04e4T + 7.83e10T^{2} \)
29 \( 1 + (1.13e6 - 3.67e5i)T + (4.04e11 - 2.94e11i)T^{2} \)
31 \( 1 + (-2.18e5 - 1.58e5i)T + (2.63e11 + 8.11e11i)T^{2} \)
37 \( 1 + (3.21e5 + 9.88e5i)T + (-2.84e12 + 2.06e12i)T^{2} \)
41 \( 1 + (3.95e6 + 1.28e6i)T + (6.45e12 + 4.69e12i)T^{2} \)
43 \( 1 + 3.56e6iT - 1.16e13T^{2} \)
47 \( 1 + (-1.62e6 + 5.01e6i)T + (-1.92e13 - 1.39e13i)T^{2} \)
53 \( 1 + (-3.07e6 - 2.23e6i)T + (1.92e13 + 5.92e13i)T^{2} \)
59 \( 1 + (-1.60e6 - 4.93e6i)T + (-1.18e14 + 8.63e13i)T^{2} \)
61 \( 1 + (-9.53e6 - 1.31e7i)T + (-5.92e13 + 1.82e14i)T^{2} \)
67 \( 1 + 1.94e7T + 4.06e14T^{2} \)
71 \( 1 + (2.30e7 - 1.67e7i)T + (1.99e14 - 6.14e14i)T^{2} \)
73 \( 1 + (-2.69e7 + 8.74e6i)T + (6.52e14 - 4.74e14i)T^{2} \)
79 \( 1 + (-3.03e7 + 4.18e7i)T + (-4.68e14 - 1.44e15i)T^{2} \)
83 \( 1 + (1.54e7 + 2.12e7i)T + (-6.95e14 + 2.14e15i)T^{2} \)
89 \( 1 + 1.14e8T + 3.93e15T^{2} \)
97 \( 1 + (-4.97e6 - 3.61e6i)T + (2.42e15 + 7.45e15i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.84340265027353683414703984507, −15.14334939638530166765803471523, −13.46701366955989008708354942540, −11.37184921590283309280320491712, −10.38497713527558201084042723774, −8.911397416305079492011571642728, −7.26041620177576009525887169476, −5.50038882325282786863633479006, −3.58942349862382125383213441932, −0.00652528005949434750485161387, 1.64432653815038360654011204137, 4.11688669088480024455962846406, 6.70045354387747119965613351058, 8.050723684677468038527134184381, 9.657709538939906952388779660233, 11.42568063432135788776854881845, 12.46940045736423608318102607246, 13.26447755555618689587875417248, 15.38170805807548308864911326065, 16.73552866880776408612007574049

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