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Properties

Label 2-11e2-11.4-c3-0-20
Degree 22
Conductor 121121
Sign −0.834+0.551i-0.834 + 0.551i
Analytic cond. 7.139237.13923
Root an. cond. 2.671932.67193
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.592 − 0.430i)2-s + (1.83 − 5.63i)3-s + (−2.30 − 7.09i)4-s + (10.4 − 7.55i)5-s + (−3.51 + 2.55i)6-s + (−5.23 − 16.0i)7-s + (−3.49 + 10.7i)8-s + (−6.58 − 4.78i)9-s − 9.41·10-s − 44.2·12-s + (60.3 + 43.8i)13-s + (−3.82 + 11.7i)14-s + (−23.5 − 72.4i)15-s + (−41.6 + 30.2i)16-s + (−66.9 + 48.6i)17-s + (1.84 + 5.66i)18-s + ⋯
L(s)  = 1  + (−0.209 − 0.152i)2-s + (0.352 − 1.08i)3-s + (−0.288 − 0.887i)4-s + (0.930 − 0.675i)5-s + (−0.238 + 0.173i)6-s + (−0.282 − 0.869i)7-s + (−0.154 + 0.475i)8-s + (−0.244 − 0.177i)9-s − 0.297·10-s − 1.06·12-s + (1.28 + 0.936i)13-s + (−0.0731 + 0.224i)14-s + (−0.405 − 1.24i)15-s + (−0.650 + 0.472i)16-s + (−0.955 + 0.694i)17-s + (0.0241 + 0.0742i)18-s + ⋯

Functional equation

Λ(s)=(121s/2 ΓC(s) L(s)=((−0.834+0.551i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 + 0.551i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(121s/2 ΓC(s+3/2) L(s)=((−0.834+0.551i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.834 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 121121    =    11211^{2}
Sign: −0.834+0.551i-0.834 + 0.551i
Analytic conductor: 7.139237.13923
Root analytic conductor: 2.671932.67193
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ121(81,⋅)\chi_{121} (81, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 121, ( :3/2), −0.834+0.551i)(2,\ 121,\ (\ :3/2),\ -0.834 + 0.551i)

Particular Values

L(2)L(2) ≈\approx 0.483353−1.60699i0.483353 - 1.60699i
L(12)L(\frac12) ≈\approx 0.483353−1.60699i0.483353 - 1.60699i
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad11 1 1
good2 1+(0.592+0.430i)T+(2.47+7.60i)T2 1 + (0.592 + 0.430i)T + (2.47 + 7.60i)T^{2}
3 1+(−1.83+5.63i)T+(−21.8−15.8i)T2 1 + (-1.83 + 5.63i)T + (-21.8 - 15.8i)T^{2}
5 1+(−10.4+7.55i)T+(38.6−118.i)T2 1 + (-10.4 + 7.55i)T + (38.6 - 118. i)T^{2}
7 1+(5.23+16.0i)T+(−277.+201.i)T2 1 + (5.23 + 16.0i)T + (-277. + 201. i)T^{2}
13 1+(−60.3−43.8i)T+(678.+2.08e3i)T2 1 + (-60.3 - 43.8i)T + (678. + 2.08e3i)T^{2}
17 1+(66.9−48.6i)T+(1.51e3−4.67e3i)T2 1 + (66.9 - 48.6i)T + (1.51e3 - 4.67e3i)T^{2}
19 1+(−20.9+64.5i)T+(−5.54e3−4.03e3i)T2 1 + (-20.9 + 64.5i)T + (-5.54e3 - 4.03e3i)T^{2}
23 1−13.3T+1.21e4T2 1 - 13.3T + 1.21e4T^{2}
29 1+(52.2+160.i)T+(−1.97e4+1.43e4i)T2 1 + (52.2 + 160. i)T + (-1.97e4 + 1.43e4i)T^{2}
31 1+(−52.9−38.4i)T+(9.20e3+2.83e4i)T2 1 + (-52.9 - 38.4i)T + (9.20e3 + 2.83e4i)T^{2}
37 1+(−12.6−38.8i)T+(−4.09e4+2.97e4i)T2 1 + (-12.6 - 38.8i)T + (-4.09e4 + 2.97e4i)T^{2}
41 1+(84.9−261.i)T+(−5.57e4−4.05e4i)T2 1 + (84.9 - 261. i)T + (-5.57e4 - 4.05e4i)T^{2}
43 1−2.28T+7.95e4T2 1 - 2.28T + 7.95e4T^{2}
47 1+(−22.2+68.3i)T+(−8.39e4−6.10e4i)T2 1 + (-22.2 + 68.3i)T + (-8.39e4 - 6.10e4i)T^{2}
53 1+(−120.−87.5i)T+(4.60e4+1.41e5i)T2 1 + (-120. - 87.5i)T + (4.60e4 + 1.41e5i)T^{2}
59 1+(−168.−518.i)T+(−1.66e5+1.20e5i)T2 1 + (-168. - 518. i)T + (-1.66e5 + 1.20e5i)T^{2}
61 1+(−81.9+59.5i)T+(7.01e4−2.15e5i)T2 1 + (-81.9 + 59.5i)T + (7.01e4 - 2.15e5i)T^{2}
67 1−411.T+3.00e5T2 1 - 411.T + 3.00e5T^{2}
71 1+(−380.+276.i)T+(1.10e5−3.40e5i)T2 1 + (-380. + 276. i)T + (1.10e5 - 3.40e5i)T^{2}
73 1+(188.+580.i)T+(−3.14e5+2.28e5i)T2 1 + (188. + 580. i)T + (-3.14e5 + 2.28e5i)T^{2}
79 1+(791.+574.i)T+(1.52e5+4.68e5i)T2 1 + (791. + 574. i)T + (1.52e5 + 4.68e5i)T^{2}
83 1+(−21.1+15.3i)T+(1.76e5−5.43e5i)T2 1 + (-21.1 + 15.3i)T + (1.76e5 - 5.43e5i)T^{2}
89 1+352.T+7.04e5T2 1 + 352.T + 7.04e5T^{2}
97 1+(685.+498.i)T+(2.82e5+8.68e5i)T2 1 + (685. + 498. i)T + (2.82e5 + 8.68e5i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−13.20520410898496871164809107532, −11.45303829981682244267681486963, −10.36277917863077648376552686221, −9.284167785916455235313035355700, −8.460140286357567836371956304483, −6.85468317717693168932049733576, −6.03035228966966989689283990652, −4.43728675649401937206551137323, −1.94489691779320467634250774902, −0.999611715066848980111535947281, 2.75605264707211758283636070842, 3.79182394136474110386627490525, 5.47732434910503826566500086936, 6.76913959756156312412051395360, 8.412448038661552997443215413524, 9.178576078287646733775331455266, 10.03269454334212561663037635800, 11.08486631489157650468041850409, 12.52060555195562115226039953699, 13.45643102500135804239462987181

Graph of the ZZ-function along the critical line