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Properties

Label 2-22-11.7-c2-0-0
Degree $2$
Conductor $22$
Sign $0.299 - 0.954i$
Analytic cond. $0.599456$
Root an. cond. $0.774245$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.831 + 1.14i)2-s + (−1.32 + 4.07i)3-s + (−0.618 − 1.90i)4-s + (6.27 − 4.56i)5-s + (−3.55 − 4.89i)6-s + (−2.67 + 0.869i)7-s + (2.68 + 0.874i)8-s + (−7.54 − 5.47i)9-s + 10.9i·10-s + (2.55 − 10.6i)11-s + 8.56·12-s + (−5.81 + 8.00i)13-s + (1.23 − 3.78i)14-s + (10.2 + 31.5i)15-s + (−3.23 + 2.35i)16-s + (−4.12 − 5.67i)17-s + ⋯
L(s)  = 1  + (−0.415 + 0.572i)2-s + (−0.440 + 1.35i)3-s + (−0.154 − 0.475i)4-s + (1.25 − 0.912i)5-s + (−0.593 − 0.816i)6-s + (−0.382 + 0.124i)7-s + (0.336 + 0.109i)8-s + (−0.838 − 0.608i)9-s + 1.09i·10-s + (0.232 − 0.972i)11-s + 0.713·12-s + (−0.447 + 0.615i)13-s + (0.0878 − 0.270i)14-s + (0.684 + 2.10i)15-s + (−0.202 + 0.146i)16-s + (−0.242 − 0.333i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $0.299 - 0.954i$
Analytic conductor: \(0.599456\)
Root analytic conductor: \(0.774245\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{22} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 22,\ (\ :1),\ 0.299 - 0.954i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.592219 + 0.434979i\)
\(L(\frac12)\) \(\approx\) \(0.592219 + 0.434979i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.831 - 1.14i)T \)
11 \( 1 + (-2.55 + 10.6i)T \)
good3 \( 1 + (1.32 - 4.07i)T + (-7.28 - 5.29i)T^{2} \)
5 \( 1 + (-6.27 + 4.56i)T + (7.72 - 23.7i)T^{2} \)
7 \( 1 + (2.67 - 0.869i)T + (39.6 - 28.8i)T^{2} \)
13 \( 1 + (5.81 - 8.00i)T + (-52.2 - 160. i)T^{2} \)
17 \( 1 + (4.12 + 5.67i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (16.8 + 5.46i)T + (292. + 212. i)T^{2} \)
23 \( 1 - 17.7T + 529T^{2} \)
29 \( 1 + (29.6 - 9.62i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (-9.04 - 6.56i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (-6.14 - 18.9i)T + (-1.10e3 + 804. i)T^{2} \)
41 \( 1 + (-76.0 - 24.7i)T + (1.35e3 + 988. i)T^{2} \)
43 \( 1 + 20.7iT - 1.84e3T^{2} \)
47 \( 1 + (8.30 - 25.5i)T + (-1.78e3 - 1.29e3i)T^{2} \)
53 \( 1 + (52.0 + 37.8i)T + (868. + 2.67e3i)T^{2} \)
59 \( 1 + (11.0 + 34.1i)T + (-2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (-30.4 - 41.9i)T + (-1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 + 17.1T + 4.48e3T^{2} \)
71 \( 1 + (25.5 - 18.5i)T + (1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (-2.48 + 0.805i)T + (4.31e3 - 3.13e3i)T^{2} \)
79 \( 1 + (-47.3 + 65.2i)T + (-1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (-31.8 - 43.8i)T + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 - 85.3T + 7.92e3T^{2} \)
97 \( 1 + (-12.3 - 8.96i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−17.46381931824561347222263917950, −16.71054751000381032601064358848, −16.03723140143429310995897309978, −14.55404255024452003948488416559, −13.13312477269217982668573542740, −11.01001411152170345359907525846, −9.612055956976493172964323382658, −8.985967301909044977999039745080, −6.11735529049354510452008930543, −4.83702689320240526826313255394, 2.16544580626431145453125438091, 6.21278158787176772724109044382, 7.39125816057870753208758534560, 9.635975148888851741055491838840, 10.87139476437832717711140737819, 12.53171252140612773360835855489, 13.25215563699413618460610677366, 14.72067824268327111700654272763, 17.20216207792425572485666403793, 17.68534890082005825160554633698

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