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Properties

Label 2-72-72.43-c4-0-39
Degree 22
Conductor 7272
Sign −0.924−0.381i-0.924 - 0.381i
Analytic cond. 7.442637.44263
Root an. cond. 2.728112.72811
Motivic weight 44
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.30 − 2.25i)2-s + (4.44 − 7.82i)3-s + (5.79 + 14.9i)4-s + (−2.53 + 1.46i)5-s + (−32.3 + 15.7i)6-s + (−39.8 − 23.0i)7-s + (14.5 − 62.3i)8-s + (−41.4 − 69.6i)9-s + (11.6 + 0.896i)10-s + (27.1 − 46.9i)11-s + (142. + 21.0i)12-s + (−174. + 100. i)13-s + (79.5 + 165. i)14-s + (0.170 + 26.3i)15-s + (−188. + 172. i)16-s − 88.7·17-s + ⋯
L(s)  = 1  + (−0.825 − 0.564i)2-s + (0.494 − 0.869i)3-s + (0.362 + 0.932i)4-s + (−0.101 + 0.0586i)5-s + (−0.898 + 0.438i)6-s + (−0.813 − 0.469i)7-s + (0.227 − 0.973i)8-s + (−0.511 − 0.859i)9-s + (0.116 + 0.00896i)10-s + (0.224 − 0.388i)11-s + (0.989 + 0.146i)12-s + (−1.03 + 0.596i)13-s + (0.405 + 0.846i)14-s + (0.000758 + 0.117i)15-s + (−0.737 + 0.675i)16-s − 0.306·17-s + ⋯

Functional equation

Λ(s)=(72s/2 ΓC(s) L(s)=((−0.924−0.381i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(72s/2 ΓC(s+2) L(s)=((−0.924−0.381i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7272    =    23⋅322^{3} \cdot 3^{2}
Sign: −0.924−0.381i-0.924 - 0.381i
Analytic conductor: 7.442637.44263
Root analytic conductor: 2.728112.72811
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ72(43,⋅)\chi_{72} (43, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 72, ( :2), −0.924−0.381i)(2,\ 72,\ (\ :2),\ -0.924 - 0.381i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.0993094+0.501181i0.0993094 + 0.501181i
L(12)L(\frac12) ≈\approx 0.0993094+0.501181i0.0993094 + 0.501181i
L(3)L(3) 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)
bad2 1+(3.30+2.25i)T 1 + (3.30 + 2.25i)T
3 1+(−4.44+7.82i)T 1 + (-4.44 + 7.82i)T
good5 1+(2.53−1.46i)T+(312.5−541.i)T2 1 + (2.53 - 1.46i)T + (312.5 - 541. i)T^{2}
7 1+(39.8+23.0i)T+(1.20e3+2.07e3i)T2 1 + (39.8 + 23.0i)T + (1.20e3 + 2.07e3i)T^{2}
11 1+(−27.1+46.9i)T+(−7.32e3−1.26e4i)T2 1 + (-27.1 + 46.9i)T + (-7.32e3 - 1.26e4i)T^{2}
13 1+(174.−100.i)T+(1.42e4−2.47e4i)T2 1 + (174. - 100. i)T + (1.42e4 - 2.47e4i)T^{2}
17 1+88.7T+8.35e4T2 1 + 88.7T + 8.35e4T^{2}
19 1+357.T+1.30e5T2 1 + 357.T + 1.30e5T^{2}
23 1+(−214.+123.i)T+(1.39e5−2.42e5i)T2 1 + (-214. + 123. i)T + (1.39e5 - 2.42e5i)T^{2}
29 1+(−18.7−10.8i)T+(3.53e5+6.12e5i)T2 1 + (-18.7 - 10.8i)T + (3.53e5 + 6.12e5i)T^{2}
31 1+(−349.+201.i)T+(4.61e5−7.99e5i)T2 1 + (-349. + 201. i)T + (4.61e5 - 7.99e5i)T^{2}
37 1+1.26e3iT−1.87e6T2 1 + 1.26e3iT - 1.87e6T^{2}
41 1+(1.10e3+1.91e3i)T+(−1.41e6+2.44e6i)T2 1 + (1.10e3 + 1.91e3i)T + (-1.41e6 + 2.44e6i)T^{2}
43 1+(−678.+1.17e3i)T+(−1.70e6−2.96e6i)T2 1 + (-678. + 1.17e3i)T + (-1.70e6 - 2.96e6i)T^{2}
47 1+(−3.12e3−1.80e3i)T+(2.43e6+4.22e6i)T2 1 + (-3.12e3 - 1.80e3i)T + (2.43e6 + 4.22e6i)T^{2}
53 1+4.12e3iT−7.89e6T2 1 + 4.12e3iT - 7.89e6T^{2}
59 1+(2.64e3+4.57e3i)T+(−6.05e6+1.04e7i)T2 1 + (2.64e3 + 4.57e3i)T + (-6.05e6 + 1.04e7i)T^{2}
61 1+(4.77e3+2.75e3i)T+(6.92e6+1.19e7i)T2 1 + (4.77e3 + 2.75e3i)T + (6.92e6 + 1.19e7i)T^{2}
67 1+(−3.09e3−5.35e3i)T+(−1.00e7+1.74e7i)T2 1 + (-3.09e3 - 5.35e3i)T + (-1.00e7 + 1.74e7i)T^{2}
71 1−4.83e3iT−2.54e7T2 1 - 4.83e3iT - 2.54e7T^{2}
73 1−9.43e3T+2.83e7T2 1 - 9.43e3T + 2.83e7T^{2}
79 1+(1.09e3+635.i)T+(1.94e7+3.37e7i)T2 1 + (1.09e3 + 635. i)T + (1.94e7 + 3.37e7i)T^{2}
83 1+(−1.26e3+2.19e3i)T+(−2.37e7−4.11e7i)T2 1 + (-1.26e3 + 2.19e3i)T + (-2.37e7 - 4.11e7i)T^{2}
89 1−5.90e3T+6.27e7T2 1 - 5.90e3T + 6.27e7T^{2}
97 1+(928.−1.60e3i)T+(−4.42e7−7.66e7i)T2 1 + (928. - 1.60e3i)T + (-4.42e7 - 7.66e7i)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

−13.01215667344183263855701562930, −12.23639244594403175389952368194, −11.04219114573874288984487732330, −9.672480853132975978593222245174, −8.750120570067924368392547167638, −7.42716575323922001532028298193, −6.58386411533717648002855747467, −3.69899731378170031217993165832, −2.20506361839364110087169081950, −0.30638900935447191618245710313, 2.60814172326957933144916997843, 4.72888810703110056546126304870, 6.24047642893998451122004939190, 7.76188750566123064653898798411, 8.931476883852904094940446201140, 9.784757360391124011688343230188, 10.64514868895051938198853061529, 12.19927788146875389881877766601, 13.75688351483183153508264478301, 15.12389910097979033451229385984

Graph of the ZZ-function along the critical line