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Properties

Label 2-242-11.10-c4-0-24
Degree $2$
Conductor $242$
Sign $0.219 - 0.975i$
Analytic cond. $25.0155$
Root an. cond. $5.00155$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82i·2-s + 16.1·3-s − 8.00·4-s + 27.1·5-s + 45.8i·6-s + 45.4i·7-s − 22.6i·8-s + 181.·9-s + 76.7i·10-s − 129.·12-s + 156. i·13-s − 128.·14-s + 439.·15-s + 64.0·16-s − 326. i·17-s + 512. i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.79·3-s − 0.500·4-s + 1.08·5-s + 1.27i·6-s + 0.926i·7-s − 0.353i·8-s + 2.23·9-s + 0.767i·10-s − 0.899·12-s + 0.926i·13-s − 0.655·14-s + 1.95·15-s + 0.250·16-s − 1.12i·17-s + 1.58i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(242\)    =    \(2 \cdot 11^{2}\)
Sign: $0.219 - 0.975i$
Analytic conductor: \(25.0155\)
Root analytic conductor: \(5.00155\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{242} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 242,\ (\ :2),\ 0.219 - 0.975i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(4.191736879\)
\(L(\frac12)\) \(\approx\) \(4.191736879\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.82iT \)
11 \( 1 \)
good3 \( 1 - 16.1T + 81T^{2} \)
5 \( 1 - 27.1T + 625T^{2} \)
7 \( 1 - 45.4iT - 2.40e3T^{2} \)
13 \( 1 - 156. iT - 2.85e4T^{2} \)
17 \( 1 + 326. iT - 8.35e4T^{2} \)
19 \( 1 - 106. iT - 1.30e5T^{2} \)
23 \( 1 + 467.T + 2.79e5T^{2} \)
29 \( 1 - 177. iT - 7.07e5T^{2} \)
31 \( 1 - 272.T + 9.23e5T^{2} \)
37 \( 1 - 2.13e3T + 1.87e6T^{2} \)
41 \( 1 + 1.29e3iT - 2.82e6T^{2} \)
43 \( 1 + 2.33e3iT - 3.41e6T^{2} \)
47 \( 1 + 2.63e3T + 4.87e6T^{2} \)
53 \( 1 - 1.63e3T + 7.89e6T^{2} \)
59 \( 1 + 4.40e3T + 1.21e7T^{2} \)
61 \( 1 - 4.75e3iT - 1.38e7T^{2} \)
67 \( 1 + 1.18e3T + 2.01e7T^{2} \)
71 \( 1 - 4.31e3T + 2.54e7T^{2} \)
73 \( 1 - 1.89e3iT - 2.83e7T^{2} \)
79 \( 1 + 1.27e3iT - 3.89e7T^{2} \)
83 \( 1 + 2.31e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.38e4T + 6.27e7T^{2} \)
97 \( 1 - 1.90e3T + 8.85e7T^{2} \)
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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

−11.93601875295268118613280768482, −10.05547342933799838030399045465, −9.359217672894512709559799797114, −8.861632359232485292747677037925, −7.86432409179790659482750352656, −6.79694332890974858158933465229, −5.61817886569236264499447394943, −4.22441119768393540445238585609, −2.74283855799204319646661799562, −1.83526131280092844134619641031, 1.28591630290826622197158312845, 2.34915863374754442673153112468, 3.40320712934619148805947540234, 4.45309643462725223670437623014, 6.20195166331843931726148115473, 7.74311547218557457716231864191, 8.352260246115375980201577948834, 9.649421696056158526401291037947, 9.917141481400510203051830168805, 10.91146817383559372368635771095

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