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Properties

Label 2-72-72.11-c3-0-4
Degree 22
Conductor 7272
Sign 0.998+0.0552i0.998 + 0.0552i
Analytic cond. 4.248134.24813
Root an. cond. 2.061102.06110
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  + (−1.80 − 2.17i)2-s + (−3.91 − 3.42i)3-s + (−1.46 + 7.86i)4-s + (7.74 + 13.4i)5-s + (−0.371 + 14.6i)6-s + (−4.08 − 2.35i)7-s + (19.7 − 11.0i)8-s + (3.59 + 26.7i)9-s + (15.1 − 41.1i)10-s + (4.32 + 2.49i)11-s + (32.6 − 25.7i)12-s + (66.4 − 38.3i)13-s + (2.25 + 13.1i)14-s + (15.6 − 79.0i)15-s + (−59.7 − 23.0i)16-s + 120. i·17-s + ⋯
L(s)  = 1  + (−0.639 − 0.769i)2-s + (−0.752 − 0.658i)3-s + (−0.182 + 0.983i)4-s + (0.693 + 1.20i)5-s + (−0.0252 + 0.999i)6-s + (−0.220 − 0.127i)7-s + (0.873 − 0.487i)8-s + (0.132 + 0.991i)9-s + (0.480 − 1.30i)10-s + (0.118 + 0.0685i)11-s + (0.784 − 0.619i)12-s + (1.41 − 0.818i)13-s + (0.0430 + 0.250i)14-s + (0.268 − 1.36i)15-s + (−0.933 − 0.359i)16-s + 1.72i·17-s + ⋯

Functional equation

Λ(s)=(72s/2 ΓC(s) L(s)=((0.998+0.0552i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0552i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(72s/2 ΓC(s+3/2) L(s)=((0.998+0.0552i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0552i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7272    =    23⋅322^{3} \cdot 3^{2}
Sign: 0.998+0.0552i0.998 + 0.0552i
Analytic conductor: 4.248134.24813
Root analytic conductor: 2.061102.06110
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ72(11,⋅)\chi_{72} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 72, ( :3/2), 0.998+0.0552i)(2,\ 72,\ (\ :3/2),\ 0.998 + 0.0552i)

Particular Values

L(2)L(2) ≈\approx 0.902292−0.0249448i0.902292 - 0.0249448i
L(12)L(\frac12) ≈\approx 0.902292−0.0249448i0.902292 - 0.0249448i
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)
bad2 1+(1.80+2.17i)T 1 + (1.80 + 2.17i)T
3 1+(3.91+3.42i)T 1 + (3.91 + 3.42i)T
good5 1+(−7.74−13.4i)T+(−62.5+108.i)T2 1 + (-7.74 - 13.4i)T + (-62.5 + 108. i)T^{2}
7 1+(4.08+2.35i)T+(171.5+297.i)T2 1 + (4.08 + 2.35i)T + (171.5 + 297. i)T^{2}
11 1+(−4.32−2.49i)T+(665.5+1.15e3i)T2 1 + (-4.32 - 2.49i)T + (665.5 + 1.15e3i)T^{2}
13 1+(−66.4+38.3i)T+(1.09e3−1.90e3i)T2 1 + (-66.4 + 38.3i)T + (1.09e3 - 1.90e3i)T^{2}
17 1−120.iT−4.91e3T2 1 - 120. iT - 4.91e3T^{2}
19 1−30.8T+6.85e3T2 1 - 30.8T + 6.85e3T^{2}
23 1+(−91.9−159.i)T+(−6.08e3+1.05e4i)T2 1 + (-91.9 - 159. i)T + (-6.08e3 + 1.05e4i)T^{2}
29 1+(−28.8+49.9i)T+(−1.21e4−2.11e4i)T2 1 + (-28.8 + 49.9i)T + (-1.21e4 - 2.11e4i)T^{2}
31 1+(37.8−21.8i)T+(1.48e4−2.57e4i)T2 1 + (37.8 - 21.8i)T + (1.48e4 - 2.57e4i)T^{2}
37 1−108.iT−5.06e4T2 1 - 108. iT - 5.06e4T^{2}
41 1+(230.−133.i)T+(3.44e4−5.96e4i)T2 1 + (230. - 133. i)T + (3.44e4 - 5.96e4i)T^{2}
43 1+(105.−181.i)T+(−3.97e4−6.88e4i)T2 1 + (105. - 181. i)T + (-3.97e4 - 6.88e4i)T^{2}
47 1+(−191.+332.i)T+(−5.19e4−8.99e4i)T2 1 + (-191. + 332. i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1−419.T+1.48e5T2 1 - 419.T + 1.48e5T^{2}
59 1+(17.1−9.91i)T+(1.02e5−1.77e5i)T2 1 + (17.1 - 9.91i)T + (1.02e5 - 1.77e5i)T^{2}
61 1+(−158.−91.4i)T+(1.13e5+1.96e5i)T2 1 + (-158. - 91.4i)T + (1.13e5 + 1.96e5i)T^{2}
67 1+(215.+373.i)T+(−1.50e5+2.60e5i)T2 1 + (215. + 373. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1+772.T+3.57e5T2 1 + 772.T + 3.57e5T^{2}
73 1+433.T+3.89e5T2 1 + 433.T + 3.89e5T^{2}
79 1+(−644.−372.i)T+(2.46e5+4.26e5i)T2 1 + (-644. - 372. i)T + (2.46e5 + 4.26e5i)T^{2}
83 1+(408.+235.i)T+(2.85e5+4.95e5i)T2 1 + (408. + 235. i)T + (2.85e5 + 4.95e5i)T^{2}
89 1−206.iT−7.04e5T2 1 - 206. iT - 7.04e5T^{2}
97 1+(−713.+1.23e3i)T+(−4.56e5−7.90e5i)T2 1 + (-713. + 1.23e3i)T + (-4.56e5 - 7.90e5i)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

−13.55295952558952452701592322398, −13.07064209235846150379263086051, −11.62180636349010140816520465424, −10.73761277252149698920135828252, −10.09691073964804319866231829844, −8.338657996349754676127543698095, −7.01583587799939749787375844818, −5.90581879505814656616080057838, −3.35122073978563504164958518563, −1.53524323135448398526695052450, 0.920384015480600605360090159118, 4.63481230646425496867790029900, 5.63777184837284400118144680563, 6.77909355345201367566808049869, 8.894460087751121125891410633770, 9.220936306144582194223931772680, 10.53690067021129966330984352147, 11.74730571715629264487219767236, 13.22347656165244026723334348791, 14.27759214656699717062522777429

Graph of the ZZ-function along the critical line