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Properties

Label 2-242-11.3-c3-0-21
Degree $2$
Conductor $242$
Sign $-0.394 + 0.918i$
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 + (1.23 + 3.80i)3-s + (1.23 − 3.80i)4-s + (−11.3 − 8.22i)5-s + (6.47 + 4.70i)6-s + (−2.47 + 7.60i)7-s + (−2.47 − 7.60i)8-s + (8.89 − 6.46i)9-s − 28·10-s + 16·12-s + (40.4 − 29.3i)13-s + (4.94 + 15.2i)14-s + (17.3 − 53.2i)15-s + (−12.9 − 9.40i)16-s + (−105. − 76.4i)17-s + (6.79 − 20.9i)18-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.237 + 0.732i)3-s + (0.154 − 0.475i)4-s + (−1.01 − 0.736i)5-s + (0.440 + 0.319i)6-s + (−0.133 + 0.410i)7-s + (−0.109 − 0.336i)8-s + (0.329 − 0.239i)9-s − 0.885·10-s + 0.384·12-s + (0.863 − 0.627i)13-s + (0.0943 + 0.290i)14-s + (0.297 − 0.916i)15-s + (−0.202 − 0.146i)16-s + (−1.50 − 1.09i)17-s + (0.0890 − 0.273i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(242\)    =    \(2 \cdot 11^{2}\)
Sign: $-0.394 + 0.918i$
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: \((2,\ 242,\ (\ :3/2),\ -0.394 + 0.918i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.921545 - 1.39875i\)
\(L(\frac12)\) \(\approx\) \(0.921545 - 1.39875i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.23 - 3.80i)T + (-21.8 + 15.8i)T^{2} \)
5 \( 1 + (11.3 + 8.22i)T + (38.6 + 118. i)T^{2} \)
7 \( 1 + (2.47 - 7.60i)T + (-277. - 201. i)T^{2} \)
13 \( 1 + (-40.4 + 29.3i)T + (678. - 2.08e3i)T^{2} \)
17 \( 1 + (105. + 76.4i)T + (1.51e3 + 4.67e3i)T^{2} \)
19 \( 1 + (33.3 + 102. i)T + (-5.54e3 + 4.03e3i)T^{2} \)
23 \( 1 + 96T + 1.21e4T^{2} \)
29 \( 1 + (-43.8 + 135. i)T + (-1.97e4 - 1.43e4i)T^{2} \)
31 \( 1 + (32.3 - 23.5i)T + (9.20e3 - 2.83e4i)T^{2} \)
37 \( 1 + (-118. + 363. i)T + (-4.09e4 - 2.97e4i)T^{2} \)
41 \( 1 + (36.4 + 112. i)T + (-5.57e4 + 4.05e4i)T^{2} \)
43 \( 1 - 220T + 7.95e4T^{2} \)
47 \( 1 + (-160. - 494. i)T + (-8.39e4 + 6.10e4i)T^{2} \)
53 \( 1 + (192. - 139. i)T + (4.60e4 - 1.41e5i)T^{2} \)
59 \( 1 + (263. - 810. i)T + (-1.66e5 - 1.20e5i)T^{2} \)
61 \( 1 + (153. + 111. i)T + (7.01e4 + 2.15e5i)T^{2} \)
67 \( 1 + 12T + 3.00e5T^{2} \)
71 \( 1 + (-90.6 - 65.8i)T + (1.10e5 + 3.40e5i)T^{2} \)
73 \( 1 + (1.85 - 5.70i)T + (-3.14e5 - 2.28e5i)T^{2} \)
79 \( 1 + (245. - 178. i)T + (1.52e5 - 4.68e5i)T^{2} \)
83 \( 1 + (663. + 481. i)T + (1.76e5 + 5.43e5i)T^{2} \)
89 \( 1 - 202T + 7.04e5T^{2} \)
97 \( 1 + (-1.13e3 + 826. i)T + (2.82e5 - 8.68e5i)T^{2} \)
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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

−11.38095875505011170697649828092, −10.65891504807883034578654774281, −9.280848775837626453908951012622, −8.791020785204160829909904130618, −7.40101117247873020266847646418, −6.00025974663896326888419195834, −4.55448926993931750348501639126, −4.12180270625787446021630105617, −2.70117718374539547296103628630, −0.53894441070383035170905213786, 1.88328283767639754539684409935, 3.59670847871327053571186721961, 4.34879524568357516876035850078, 6.29492929438826540504998434006, 6.85347838884928205803773413064, 7.87917157047199845506843814883, 8.523315667820189677095621954157, 10.34367003141498323158327474463, 11.16575377667229528082341685972, 12.12631681143106136400758352250

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