Properties

Label 2-242-11.3-c3-0-15
Degree $2$
Conductor $242$
Sign $0.0694 + 0.997i$
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 + (−1.33 − 4.09i)3-s + (1.23 − 3.80i)4-s + (6.52 + 4.73i)5-s + (6.97 + 5.06i)6-s + (8.05 − 24.8i)7-s + (2.47 + 7.60i)8-s + (6.82 − 4.95i)9-s − 16.1·10-s − 17.2·12-s + (−2.64 + 1.91i)13-s + (16.1 + 49.6i)14-s + (10.7 − 33.0i)15-s + (−12.9 − 9.40i)16-s + (−16.8 − 12.2i)17-s + (−5.21 + 16.0i)18-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.256 − 0.788i)3-s + (0.154 − 0.475i)4-s + (0.583 + 0.423i)5-s + (0.474 + 0.344i)6-s + (0.435 − 1.33i)7-s + (0.109 + 0.336i)8-s + (0.252 − 0.183i)9-s − 0.509·10-s − 0.414·12-s + (−0.0563 + 0.0409i)13-s + (0.307 + 0.946i)14-s + (0.184 − 0.568i)15-s + (−0.202 − 0.146i)16-s + (−0.240 − 0.175i)17-s + (−0.0682 + 0.210i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(242\)    =    \(2 \cdot 11^{2}\)
Sign: $0.0694 + 0.997i$
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: \((2,\ 242,\ (\ :3/2),\ 0.0694 + 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.970541 - 0.905298i\)
\(L(\frac12)\) \(\approx\) \(0.970541 - 0.905298i\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.33 + 4.09i)T + (-21.8 + 15.8i)T^{2} \)
5 \( 1 + (-6.52 - 4.73i)T + (38.6 + 118. i)T^{2} \)
7 \( 1 + (-8.05 + 24.8i)T + (-277. - 201. i)T^{2} \)
13 \( 1 + (2.64 - 1.91i)T + (678. - 2.08e3i)T^{2} \)
17 \( 1 + (16.8 + 12.2i)T + (1.51e3 + 4.67e3i)T^{2} \)
19 \( 1 + (-38.9 - 119. i)T + (-5.54e3 + 4.03e3i)T^{2} \)
23 \( 1 - 97.8T + 1.21e4T^{2} \)
29 \( 1 + (-81.5 + 250. i)T + (-1.97e4 - 1.43e4i)T^{2} \)
31 \( 1 + (161. - 117. i)T + (9.20e3 - 2.83e4i)T^{2} \)
37 \( 1 + (-112. + 347. i)T + (-4.09e4 - 2.97e4i)T^{2} \)
41 \( 1 + (84.5 + 260. i)T + (-5.57e4 + 4.05e4i)T^{2} \)
43 \( 1 + 388.T + 7.95e4T^{2} \)
47 \( 1 + (16.0 + 49.3i)T + (-8.39e4 + 6.10e4i)T^{2} \)
53 \( 1 + (-333. + 242. i)T + (4.60e4 - 1.41e5i)T^{2} \)
59 \( 1 + (-8.12 + 24.9i)T + (-1.66e5 - 1.20e5i)T^{2} \)
61 \( 1 + (-132. - 96.4i)T + (7.01e4 + 2.15e5i)T^{2} \)
67 \( 1 - 276.T + 3.00e5T^{2} \)
71 \( 1 + (-418. - 303. i)T + (1.10e5 + 3.40e5i)T^{2} \)
73 \( 1 + (-74.6 + 229. i)T + (-3.14e5 - 2.28e5i)T^{2} \)
79 \( 1 + (220. - 160. i)T + (1.52e5 - 4.68e5i)T^{2} \)
83 \( 1 + (-58.7 - 42.6i)T + (1.76e5 + 5.43e5i)T^{2} \)
89 \( 1 + 1.19e3T + 7.04e5T^{2} \)
97 \( 1 + (1.18e3 - 860. i)T + (2.82e5 - 8.68e5i)T^{2} \)
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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

−11.28136036071746395023682799114, −10.33092030505900234217095132163, −9.689946152328763568869137716716, −8.206290050005100412714193517852, −7.29608176980966985067200603442, −6.70869807314943789682242285006, −5.59338055178399292380537236114, −3.97940451019460234412269786095, −1.90598950028694218039775997549, −0.70558055547659924263823878325, 1.57399154039191841147324058917, 2.95441782250111880581647936573, 4.76893174015539985555313390781, 5.41472582863900529102604478184, 6.95253043673391322778329208786, 8.420706410173063327889764514634, 9.182280627798918611378811316374, 9.804820333025851103796833702686, 10.98581877184646551641834345256, 11.56748672218912711127682624837

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