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Properties

Label 2-242-11.3-c3-0-13
Degree $2$
Conductor $242$
Sign $0.859 - 0.511i$
Analytic cond. $14.2784$
Root an. cond. $3.77868$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.61 + 1.17i)2-s + (2.34 + 7.20i)3-s + (1.23 − 3.80i)4-s + (−4.37 − 3.17i)5-s + (−12.2 − 8.90i)6-s + (6.84 − 21.0i)7-s + (2.47 + 7.60i)8-s + (−24.5 + 17.8i)9-s + 10.8·10-s + 30.3·12-s + (62.1 − 45.1i)13-s + (13.6 + 42.1i)14-s + (12.6 − 38.9i)15-s + (−12.9 − 9.40i)16-s + (47.9 + 34.8i)17-s + (18.7 − 57.8i)18-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.450 + 1.38i)3-s + (0.154 − 0.475i)4-s + (−0.391 − 0.284i)5-s + (−0.834 − 0.605i)6-s + (0.369 − 1.13i)7-s + (0.109 + 0.336i)8-s + (−0.910 + 0.661i)9-s + 0.342·10-s + 0.728·12-s + (1.32 − 0.963i)13-s + (0.261 + 0.804i)14-s + (0.217 − 0.670i)15-s + (−0.202 − 0.146i)16-s + (0.684 + 0.497i)17-s + (0.245 − 0.756i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(242\)    =    \(2 \cdot 11^{2}\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(14.2784\)
Root analytic conductor: \(3.77868\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{242} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 242,\ (\ :3/2),\ 0.859 - 0.511i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.50815 + 0.414796i\)
\(L(\frac12)\) \(\approx\) \(1.50815 + 0.414796i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.61 - 1.17i)T \)
11 \( 1 \)
good3 \( 1 + (-2.34 - 7.20i)T + (-21.8 + 15.8i)T^{2} \)
5 \( 1 + (4.37 + 3.17i)T + (38.6 + 118. i)T^{2} \)
7 \( 1 + (-6.84 + 21.0i)T + (-277. - 201. i)T^{2} \)
13 \( 1 + (-62.1 + 45.1i)T + (678. - 2.08e3i)T^{2} \)
17 \( 1 + (-47.9 - 34.8i)T + (1.51e3 + 4.67e3i)T^{2} \)
19 \( 1 + (29.4 + 90.6i)T + (-5.54e3 + 4.03e3i)T^{2} \)
23 \( 1 - 142.T + 1.21e4T^{2} \)
29 \( 1 + (-6.30 + 19.4i)T + (-1.97e4 - 1.43e4i)T^{2} \)
31 \( 1 + (172. - 125. i)T + (9.20e3 - 2.83e4i)T^{2} \)
37 \( 1 + (44.9 - 138. i)T + (-4.09e4 - 2.97e4i)T^{2} \)
41 \( 1 + (25.5 + 78.7i)T + (-5.57e4 + 4.05e4i)T^{2} \)
43 \( 1 - 151.T + 7.95e4T^{2} \)
47 \( 1 + (-27.9 - 85.9i)T + (-8.39e4 + 6.10e4i)T^{2} \)
53 \( 1 + (-189. + 138. i)T + (4.60e4 - 1.41e5i)T^{2} \)
59 \( 1 + (-93.4 + 287. i)T + (-1.66e5 - 1.20e5i)T^{2} \)
61 \( 1 + (120. + 87.7i)T + (7.01e4 + 2.15e5i)T^{2} \)
67 \( 1 - 826.T + 3.00e5T^{2} \)
71 \( 1 + (-727. - 528. i)T + (1.10e5 + 3.40e5i)T^{2} \)
73 \( 1 + (-42.6 + 131. i)T + (-3.14e5 - 2.28e5i)T^{2} \)
79 \( 1 + (246. - 178. i)T + (1.52e5 - 4.68e5i)T^{2} \)
83 \( 1 + (618. + 449. i)T + (1.76e5 + 5.43e5i)T^{2} \)
89 \( 1 + 313.T + 7.04e5T^{2} \)
97 \( 1 + (-471. + 342. i)T + (2.82e5 - 8.68e5i)T^{2} \)
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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

−11.04345093804489734093113449953, −10.74464394703589152188862885977, −9.828747826222185624204245356911, −8.736917391073638532780883596362, −8.156955697509921051489273640104, −6.93942031478294053339375384383, −5.37836061917865820842014618881, −4.30883268347842870161288171599, −3.34025798147352351431329175783, −0.874003714192287645392728151918, 1.27718319889086733295405123995, 2.28604333210652822536813127408, 3.61448503216742722832033898624, 5.70995364398750343340865859541, 6.86930178441194735512189418595, 7.75932921682175583587328768428, 8.598306949861141712857292628530, 9.296042305855302697489671918694, 10.94010189748519459427532610073, 11.66722090245755262465088857088

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