Properties

Label 2-22-11.9-c11-0-2
Degree $2$
Conductor $22$
Sign $0.866 - 0.499i$
Analytic cond. $16.9035$
Root an. cond. $4.11139$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (9.88 − 30.4i)2-s + (−507. + 369. i)3-s + (−828. − 601. i)4-s + (−2.31e3 − 7.13e3i)5-s + (6.20e3 + 1.91e4i)6-s + (−5.20e4 − 3.78e4i)7-s + (−2.65e4 + 1.92e4i)8-s + (6.70e4 − 2.06e5i)9-s − 2.40e5·10-s + (4.77e5 + 2.39e5i)11-s + 6.42e5·12-s + (−5.94e5 + 1.82e6i)13-s + (−1.66e6 + 1.20e6i)14-s + (3.81e6 + 2.76e6i)15-s + (3.24e5 + 9.97e5i)16-s + (8.65e5 + 2.66e6i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−1.20 + 0.876i)3-s + (−0.404 − 0.293i)4-s + (−0.331 − 1.02i)5-s + (0.325 + 1.00i)6-s + (−1.17 − 0.850i)7-s + (−0.286 + 0.207i)8-s + (0.378 − 1.16i)9-s − 0.759·10-s + (0.893 + 0.448i)11-s + 0.745·12-s + (−0.443 + 1.36i)13-s + (−0.827 + 0.601i)14-s + (1.29 + 0.941i)15-s + (0.0772 + 0.237i)16-s + (0.147 + 0.455i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $0.866 - 0.499i$
Analytic conductor: \(16.9035\)
Root analytic conductor: \(4.11139\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{22} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 22,\ (\ :11/2),\ 0.866 - 0.499i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.652921 + 0.174792i\)
\(L(\frac12)\) \(\approx\) \(0.652921 + 0.174792i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-9.88 + 30.4i)T \)
11 \( 1 + (-4.77e5 - 2.39e5i)T \)
good3 \( 1 + (507. - 369. i)T + (5.47e4 - 1.68e5i)T^{2} \)
5 \( 1 + (2.31e3 + 7.13e3i)T + (-3.95e7 + 2.87e7i)T^{2} \)
7 \( 1 + (5.20e4 + 3.78e4i)T + (6.11e8 + 1.88e9i)T^{2} \)
13 \( 1 + (5.94e5 - 1.82e6i)T + (-1.44e12 - 1.05e12i)T^{2} \)
17 \( 1 + (-8.65e5 - 2.66e6i)T + (-2.77e13 + 2.01e13i)T^{2} \)
19 \( 1 + (-1.34e7 + 9.76e6i)T + (3.59e13 - 1.10e14i)T^{2} \)
23 \( 1 + 5.10e7T + 9.52e14T^{2} \)
29 \( 1 + (-1.71e8 - 1.24e8i)T + (3.77e15 + 1.16e16i)T^{2} \)
31 \( 1 + (5.45e6 - 1.67e7i)T + (-2.05e16 - 1.49e16i)T^{2} \)
37 \( 1 + (-1.07e8 - 7.83e7i)T + (5.49e16 + 1.69e17i)T^{2} \)
41 \( 1 + (-1.63e8 + 1.18e8i)T + (1.70e17 - 5.23e17i)T^{2} \)
43 \( 1 + 7.57e7T + 9.29e17T^{2} \)
47 \( 1 + (5.46e8 - 3.97e8i)T + (7.63e17 - 2.35e18i)T^{2} \)
53 \( 1 + (-2.29e8 + 7.05e8i)T + (-7.49e18 - 5.44e18i)T^{2} \)
59 \( 1 + (3.91e9 + 2.84e9i)T + (9.31e18 + 2.86e19i)T^{2} \)
61 \( 1 + (-1.69e9 - 5.22e9i)T + (-3.52e19 + 2.55e19i)T^{2} \)
67 \( 1 + 2.59e9T + 1.22e20T^{2} \)
71 \( 1 + (-5.78e9 - 1.78e10i)T + (-1.86e20 + 1.35e20i)T^{2} \)
73 \( 1 + (-2.10e9 - 1.53e9i)T + (9.69e19 + 2.98e20i)T^{2} \)
79 \( 1 + (-7.35e9 + 2.26e10i)T + (-6.05e20 - 4.39e20i)T^{2} \)
83 \( 1 + (-1.16e10 - 3.59e10i)T + (-1.04e21 + 7.56e20i)T^{2} \)
89 \( 1 - 2.38e8T + 2.77e21T^{2} \)
97 \( 1 + (3.47e10 - 1.07e11i)T + (-5.78e21 - 4.20e21i)T^{2} \)
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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

−16.04572464109136900368181789739, −13.98593431963521645862379781180, −12.40149259720133572434953112752, −11.67758100399682451123532193116, −10.15893643583028677684597142588, −9.303039888190596972209303616865, −6.57088109002260735181221243564, −4.79430934806051813499109307739, −3.92933861540801043824957613808, −0.928912725212554537777961761763, 0.39268009689155710571394071046, 3.14498807741844225831053045842, 5.76490848035555312747579335580, 6.42337639217353319901634949694, 7.71942412247060939447931355873, 9.955399395823884011609221943233, 11.75630866900635482474722862568, 12.45474066388659209431310837823, 13.98387956776093946536973511072, 15.47743264057990625524893788235

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