Properties

Label 2-22-11.2-c8-0-6
Degree $2$
Conductor $22$
Sign $-0.990 + 0.137i$
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  + (−10.7 − 3.49i)2-s + (−9.77 + 7.10i)3-s + (103. + 75.2i)4-s + (24.4 + 75.2i)5-s + (130. − 42.2i)6-s + (1.39e3 − 1.92e3i)7-s + (−851. − 1.17e3i)8-s + (−1.98e3 + 6.10e3i)9-s − 895. i·10-s + (−1.30e4 − 6.60e3i)11-s − 1.54e3·12-s + (−4.30e4 − 1.39e4i)13-s + (−2.17e4 + 1.57e4i)14-s + (−773. − 562. i)15-s + (5.06e3 + 1.55e4i)16-s + (−9.44e4 + 3.06e4i)17-s + ⋯
L(s)  = 1  + (−0.672 − 0.218i)2-s + (−0.120 + 0.0877i)3-s + (0.404 + 0.293i)4-s + (0.0391 + 0.120i)5-s + (0.100 − 0.0326i)6-s + (0.581 − 0.800i)7-s + (−0.207 − 0.286i)8-s + (−0.302 + 0.929i)9-s − 0.0895i·10-s + (−0.892 − 0.451i)11-s − 0.0746·12-s + (−1.50 − 0.489i)13-s + (−0.565 + 0.411i)14-s + (−0.0152 − 0.0111i)15-s + (0.0772 + 0.237i)16-s + (−1.13 + 0.367i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.137i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $-0.990 + 0.137i$
Analytic conductor: \(8.96232\)
Root analytic conductor: \(2.99371\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{22} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 22,\ (\ :4),\ -0.990 + 0.137i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.0135369 - 0.195590i\)
\(L(\frac12)\) \(\approx\) \(0.0135369 - 0.195590i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (10.7 + 3.49i)T \)
11 \( 1 + (1.30e4 + 6.60e3i)T \)
good3 \( 1 + (9.77 - 7.10i)T + (2.02e3 - 6.23e3i)T^{2} \)
5 \( 1 + (-24.4 - 75.2i)T + (-3.16e5 + 2.29e5i)T^{2} \)
7 \( 1 + (-1.39e3 + 1.92e3i)T + (-1.78e6 - 5.48e6i)T^{2} \)
13 \( 1 + (4.30e4 + 1.39e4i)T + (6.59e8 + 4.79e8i)T^{2} \)
17 \( 1 + (9.44e4 - 3.06e4i)T + (5.64e9 - 4.10e9i)T^{2} \)
19 \( 1 + (6.08e4 + 8.37e4i)T + (-5.24e9 + 1.61e10i)T^{2} \)
23 \( 1 + 46.8T + 7.83e10T^{2} \)
29 \( 1 + (-8.11e4 + 1.11e5i)T + (-1.54e11 - 4.75e11i)T^{2} \)
31 \( 1 + (2.98e5 - 9.17e5i)T + (-6.90e11 - 5.01e11i)T^{2} \)
37 \( 1 + (2.76e6 + 2.00e6i)T + (1.08e12 + 3.34e12i)T^{2} \)
41 \( 1 + (-1.08e6 - 1.49e6i)T + (-2.46e12 + 7.59e12i)T^{2} \)
43 \( 1 + 2.14e6iT - 1.16e13T^{2} \)
47 \( 1 + (3.87e6 - 2.81e6i)T + (7.35e12 - 2.26e13i)T^{2} \)
53 \( 1 + (6.52e5 - 2.00e6i)T + (-5.03e13 - 3.65e13i)T^{2} \)
59 \( 1 + (-6.45e6 - 4.69e6i)T + (4.53e13 + 1.39e14i)T^{2} \)
61 \( 1 + (-1.05e7 + 3.42e6i)T + (1.55e14 - 1.12e14i)T^{2} \)
67 \( 1 - 2.72e7T + 4.06e14T^{2} \)
71 \( 1 + (-5.43e6 - 1.67e7i)T + (-5.22e14 + 3.79e14i)T^{2} \)
73 \( 1 + (-1.95e7 + 2.68e7i)T + (-2.49e14 - 7.66e14i)T^{2} \)
79 \( 1 + (5.01e7 + 1.63e7i)T + (1.22e15 + 8.91e14i)T^{2} \)
83 \( 1 + (-1.12e7 + 3.64e6i)T + (1.82e15 - 1.32e15i)T^{2} \)
89 \( 1 + 8.25e7T + 3.93e15T^{2} \)
97 \( 1 + (-2.70e7 + 8.32e7i)T + (-6.34e15 - 4.60e15i)T^{2} \)
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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

−15.79288824826790812508553667291, −14.25242527292101386487398965102, −12.83212278519681166779369784645, −11.02894713266210067882507122325, −10.37553422924117080076427094689, −8.447620537239882000845592711614, −7.22622864013202214653777081527, −4.91307339851610228942158422424, −2.40963598312272065798724111770, −0.11127895010245027551481914901, 2.22810596300966080990601809105, 5.14605006359258153821906640711, 6.92987229065773995636264102431, 8.534031431657126179353067167302, 9.760658776246770499249315501191, 11.44727145651606354800215245823, 12.57680860482028773043549354454, 14.62931697762508441127476641840, 15.42958770335092833653704946534, 17.00993521046892306004600869782

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