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Properties

Label 2-72-8.3-c2-0-1
Degree 22
Conductor 7272
Sign 0.774−0.632i0.774 - 0.632i
Analytic cond. 1.961851.96185
Root an. cond. 1.400661.40066
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 − 1.46i)2-s + (−0.267 + 3.99i)4-s + 7.98i·5-s + 2.13i·7-s + (6.19 − 5.06i)8-s + (11.6 − 10.9i)10-s + 8·11-s + 11.6i·13-s + (3.12 − 2.92i)14-s + (−15.8 − 2.13i)16-s − 11.8·17-s + 14.9·19-s + (−31.8 − 2.13i)20-s + (−10.9 − 11.6i)22-s + 4.27i·23-s + ⋯
L(s)  = 1  + (−0.683 − 0.730i)2-s + (−0.0669 + 0.997i)4-s + 1.59i·5-s + 0.305i·7-s + (0.774 − 0.632i)8-s + (1.16 − 1.09i)10-s + 0.727·11-s + 0.898i·13-s + (0.223 − 0.208i)14-s + (−0.991 − 0.133i)16-s − 0.697·17-s + 0.785·19-s + (−1.59 − 0.106i)20-s + (−0.496 − 0.531i)22-s + 0.185i·23-s + ⋯

Functional equation

Λ(s)=(72s/2 ΓC(s) L(s)=((0.774−0.632i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(72s/2 ΓC(s+1) L(s)=((0.774−0.632i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7272    =    23⋅322^{3} \cdot 3^{2}
Sign: 0.774−0.632i0.774 - 0.632i
Analytic conductor: 1.961851.96185
Root analytic conductor: 1.400661.40066
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ72(19,⋅)\chi_{72} (19, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 72, ( :1), 0.774−0.632i)(2,\ 72,\ (\ :1),\ 0.774 - 0.632i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.809769+0.288653i0.809769 + 0.288653i
L(12)L(\frac12) ≈\approx 0.809769+0.288653i0.809769 + 0.288653i
L(2)L(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+(1.36+1.46i)T 1 + (1.36 + 1.46i)T
3 1 1
good5 1−7.98iT−25T2 1 - 7.98iT - 25T^{2}
7 1−2.13iT−49T2 1 - 2.13iT - 49T^{2}
11 1−8T+121T2 1 - 8T + 121T^{2}
13 1−11.6iT−169T2 1 - 11.6iT - 169T^{2}
17 1+11.8T+289T2 1 + 11.8T + 289T^{2}
19 1−14.9T+361T2 1 - 14.9T + 361T^{2}
23 1−4.27iT−529T2 1 - 4.27iT - 529T^{2}
29 1+0.573iT−841T2 1 + 0.573iT - 841T^{2}
31 1+57.4iT−961T2 1 + 57.4iT - 961T^{2}
37 1−27.6iT−1.36e3T2 1 - 27.6iT - 1.36e3T^{2}
41 1−31.5T+1.68e3T2 1 - 31.5T + 1.68e3T^{2}
43 1−28.7T+1.84e3T2 1 - 28.7T + 1.84e3T^{2}
47 1+59.5iT−2.20e3T2 1 + 59.5iT - 2.20e3T^{2}
53 1+31.3iT−2.80e3T2 1 + 31.3iT - 2.80e3T^{2}
59 1−52.7T+3.48e3T2 1 - 52.7T + 3.48e3T^{2}
61 1+59.5iT−3.72e3T2 1 + 59.5iT - 3.72e3T^{2}
67 1+84.7T+4.48e3T2 1 + 84.7T + 4.48e3T^{2}
71 1−42.4iT−5.04e3T2 1 - 42.4iT - 5.04e3T^{2}
73 1+5.42T+5.32e3T2 1 + 5.42T + 5.32e3T^{2}
79 1−44.6iT−6.24e3T2 1 - 44.6iT - 6.24e3T^{2}
83 1+67.7T+6.88e3T2 1 + 67.7T + 6.88e3T^{2}
89 1−133.T+7.92e3T2 1 - 133.T + 7.92e3T^{2}
97 1−97.1T+9.40e3T2 1 - 97.1T + 9.40e3T^{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

−14.43244951597317680512210083523, −13.41475164075039643012502703397, −11.76139397506278485834806280415, −11.27984212606822627611752612148, −10.06810863887070178928168700290, −9.080282463822533586893818136635, −7.49800902505051585275999164462, −6.48455552306572344912327671009, −3.86097316195256239732338945321, −2.36620447646403876290220758775, 1.03992153995220372661567779604, 4.57227852994403919594246230971, 5.77324905175859997564278806102, 7.39061357997135212246354669911, 8.622317305641944247219840581777, 9.318578485766785567438175810599, 10.66160014150586465012153707820, 12.16612926128509256214956913106, 13.29531537388394532190326164353, 14.38363123878087061069384512078

Graph of the ZZ-function along the critical line