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Properties

Label 2-7200-120.59-c1-0-69
Degree 22
Conductor 72007200
Sign −0.838+0.544i-0.838 + 0.544i
Analytic cond. 57.492257.4922
Root an. cond. 7.582367.58236
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·7-s − 6.37i·11-s − 3.54·13-s + 3.92·17-s + 1.27·19-s + 6.28i·23-s − 9.00·29-s − 3.92i·31-s + 2.51·37-s − 5.27i·41-s − 1.55i·43-s − 9.73i·47-s − 5·49-s − 5.55i·53-s + 0.313i·59-s + ⋯
L(s)  = 1  + 0.534·7-s − 1.92i·11-s − 0.982·13-s + 0.952·17-s + 0.292·19-s + 1.31i·23-s − 1.67·29-s − 0.705i·31-s + 0.413·37-s − 0.823i·41-s − 0.237i·43-s − 1.42i·47-s − 0.714·49-s − 0.763i·53-s + 0.0408i·59-s + ⋯

Functional equation

Λ(s)=(7200s/2 ΓC(s) L(s)=((−0.838+0.544i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.838 + 0.544i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(7200s/2 ΓC(s+1/2) L(s)=((−0.838+0.544i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.838 + 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 72007200    =    25⋅32⋅522^{5} \cdot 3^{2} \cdot 5^{2}
Sign: −0.838+0.544i-0.838 + 0.544i
Analytic conductor: 57.492257.4922
Root analytic conductor: 7.582367.58236
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ7200(3599,⋅)\chi_{7200} (3599, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 7200, ( :1/2), −0.838+0.544i)(2,\ 7200,\ (\ :1/2),\ -0.838 + 0.544i)

Particular Values

L(1)L(1) ≈\approx 1.0124749961.012474996
L(12)L(\frac12) ≈\approx 1.0124749961.012474996
L(32)L(\frac{3}{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
3 1 1
5 1 1
good7 1−1.41T+7T2 1 - 1.41T + 7T^{2}
11 1+6.37iT−11T2 1 + 6.37iT - 11T^{2}
13 1+3.54T+13T2 1 + 3.54T + 13T^{2}
17 1−3.92T+17T2 1 - 3.92T + 17T^{2}
19 1−1.27T+19T2 1 - 1.27T + 19T^{2}
23 1−6.28iT−23T2 1 - 6.28iT - 23T^{2}
29 1+9.00T+29T2 1 + 9.00T + 29T^{2}
31 1+3.92iT−31T2 1 + 3.92iT - 31T^{2}
37 1−2.51T+37T2 1 - 2.51T + 37T^{2}
41 1+5.27iT−41T2 1 + 5.27iT - 41T^{2}
43 1+1.55iT−43T2 1 + 1.55iT - 43T^{2}
47 1+9.73iT−47T2 1 + 9.73iT - 47T^{2}
53 1+5.55iT−53T2 1 + 5.55iT - 53T^{2}
59 1−0.313iT−59T2 1 - 0.313iT - 59T^{2}
61 1−12.7iT−61T2 1 - 12.7iT - 61T^{2}
67 1+7.00iT−67T2 1 + 7.00iT - 67T^{2}
71 1+0.990T+71T2 1 + 0.990T + 71T^{2}
73 1−12.0iT−73T2 1 - 12.0iT - 73T^{2}
79 1+8.18iT−79T2 1 + 8.18iT - 79T^{2}
83 1+5.02T+83T2 1 + 5.02T + 83T^{2}
89 1+0.386iT−89T2 1 + 0.386iT - 89T^{2}
97 1+10.4iT−97T2 1 + 10.4iT - 97T^{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

−7.59976917008668856728024355176, −7.18646486890477842516667880505, −5.96801134965998878044767882472, −5.59612893625230404154037352485, −5.00186604125565790859909658345, −3.79060508815061752600663673080, −3.38231145964530390355014428722, −2.35974601817374221999171781179, −1.33046794068446672422995208282, −0.24169496041735158196782081601, 1.34410274441665682760457962129, 2.13679070038538337547645172183, 2.93911781290695069822126790080, 4.05820395799073134890072573119, 4.78590309444782465126389518645, 5.12262719877715450656029326381, 6.17593399073173064947368204412, 6.94168031148142188020376805656, 7.67889241067010748494950964028, 7.83948804100899595177777869835

Graph of the ZZ-function along the critical line