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Properties

Label 2-189-7.2-c3-0-9
Degree 22
Conductor 189189
Sign −0.550−0.834i-0.550 - 0.834i
Analytic cond. 11.151311.1513
Root an. cond. 3.339363.33936
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  + (−2.73 + 4.73i)2-s + (−10.9 − 18.9i)4-s + (−0.0995 + 0.172i)5-s + (−16.0 − 9.31i)7-s + 75.5·8-s + (−0.543 − 0.941i)10-s + (−14.2 − 24.6i)11-s + 32.5·13-s + (87.7 − 50.2i)14-s + (−118. + 206. i)16-s + (57.7 + 100. i)17-s + (−10.5 + 18.3i)19-s + 4.34·20-s + 155.·22-s + (−46.8 + 81.2i)23-s + ⋯
L(s)  = 1  + (−0.965 + 1.67i)2-s + (−1.36 − 2.36i)4-s + (−0.00890 + 0.0154i)5-s + (−0.864 − 0.503i)7-s + 3.33·8-s + (−0.0171 − 0.0297i)10-s + (−0.389 − 0.674i)11-s + 0.695·13-s + (1.67 − 0.959i)14-s + (−1.85 + 3.21i)16-s + (0.823 + 1.42i)17-s + (−0.127 + 0.221i)19-s + 0.0486·20-s + 1.50·22-s + (−0.425 + 0.736i)23-s + ⋯

Functional equation

Λ(s)=(189s/2 ΓC(s) L(s)=((−0.550−0.834i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.550 - 0.834i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(189s/2 ΓC(s+3/2) L(s)=((−0.550−0.834i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 189189    =    33⋅73^{3} \cdot 7
Sign: −0.550−0.834i-0.550 - 0.834i
Analytic conductor: 11.151311.1513
Root analytic conductor: 3.339363.33936
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ189(163,⋅)\chi_{189} (163, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 189, ( :3/2), −0.550−0.834i)(2,\ 189,\ (\ :3/2),\ -0.550 - 0.834i)

Particular Values

L(2)L(2) ≈\approx 0.357791+0.664884i0.357791 + 0.664884i
L(12)L(\frac12) ≈\approx 0.357791+0.664884i0.357791 + 0.664884i
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)
bad3 1 1
7 1+(16.0+9.31i)T 1 + (16.0 + 9.31i)T
good2 1+(2.73−4.73i)T+(−4−6.92i)T2 1 + (2.73 - 4.73i)T + (-4 - 6.92i)T^{2}
5 1+(0.0995−0.172i)T+(−62.5−108.i)T2 1 + (0.0995 - 0.172i)T + (-62.5 - 108. i)T^{2}
11 1+(14.2+24.6i)T+(−665.5+1.15e3i)T2 1 + (14.2 + 24.6i)T + (-665.5 + 1.15e3i)T^{2}
13 1−32.5T+2.19e3T2 1 - 32.5T + 2.19e3T^{2}
17 1+(−57.7−100.i)T+(−2.45e3+4.25e3i)T2 1 + (-57.7 - 100. i)T + (-2.45e3 + 4.25e3i)T^{2}
19 1+(10.5−18.3i)T+(−3.42e3−5.94e3i)T2 1 + (10.5 - 18.3i)T + (-3.42e3 - 5.94e3i)T^{2}
23 1+(46.8−81.2i)T+(−6.08e3−1.05e4i)T2 1 + (46.8 - 81.2i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1−231.T+2.43e4T2 1 - 231.T + 2.43e4T^{2}
31 1+(140.+243.i)T+(−1.48e4+2.57e4i)T2 1 + (140. + 243. i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1+(−73.2+126.i)T+(−2.53e4−4.38e4i)T2 1 + (-73.2 + 126. i)T + (-2.53e4 - 4.38e4i)T^{2}
41 1+111.T+6.89e4T2 1 + 111.T + 6.89e4T^{2}
43 1−392.T+7.95e4T2 1 - 392.T + 7.95e4T^{2}
47 1+(136.−236.i)T+(−5.19e4−8.99e4i)T2 1 + (136. - 236. i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1+(−170.−294.i)T+(−7.44e4+1.28e5i)T2 1 + (-170. - 294. i)T + (-7.44e4 + 1.28e5i)T^{2}
59 1+(−348.−603.i)T+(−1.02e5+1.77e5i)T2 1 + (-348. - 603. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(−185.+320.i)T+(−1.13e5−1.96e5i)T2 1 + (-185. + 320. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(43.5+75.4i)T+(−1.50e5+2.60e5i)T2 1 + (43.5 + 75.4i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1+88.3T+3.57e5T2 1 + 88.3T + 3.57e5T^{2}
73 1+(−401.−695.i)T+(−1.94e5+3.36e5i)T2 1 + (-401. - 695. i)T + (-1.94e5 + 3.36e5i)T^{2}
79 1+(182.−315.i)T+(−2.46e5−4.26e5i)T2 1 + (182. - 315. i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1−921.T+5.71e5T2 1 - 921.T + 5.71e5T^{2}
89 1+(−105.+183.i)T+(−3.52e5−6.10e5i)T2 1 + (-105. + 183. i)T + (-3.52e5 - 6.10e5i)T^{2}
97 1−845.T+9.12e5T2 1 - 845.T + 9.12e5T^{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

−12.87735576177266559093735021616, −10.89680082827782186922795659187, −10.15636881542475604241666389550, −9.207382601688159548893463223858, −8.223084439362166806767897501558, −7.42877386335435633841461829768, −6.22784683997926844683193028348, −5.67204312652271760719729679775, −3.90311505108765299014561907436, −0.965780004949306879514919697094, 0.64346038684162800878137676299, 2.39497413696721882004309642221, 3.34217234789234225204990978917, 4.83679382966102535207946364842, 6.87905317277454545791515780409, 8.218660157973910262449816696402, 9.054856604802664092198099525786, 9.969001163177901794789942188506, 10.57075777011696155496618988141, 11.84242167262619989168104412853

Graph of the ZZ-function along the critical line