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Properties

Label 2-7200-5.4-c1-0-36
Degree 22
Conductor 72007200
Sign 0.447−0.894i0.447 - 0.894i
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  + 3i·7-s + 5i·13-s + 5·19-s − 4i·23-s + 4·29-s + 5·31-s − 10i·37-s + 10·41-s + i·43-s − 2i·47-s − 2·49-s − 10i·53-s + 10·59-s − 5·61-s + 3i·67-s + ⋯
L(s)  = 1  + 1.13i·7-s + 1.38i·13-s + 1.14·19-s − 0.834i·23-s + 0.742·29-s + 0.898·31-s − 1.64i·37-s + 1.56·41-s + 0.152i·43-s − 0.291i·47-s − 0.285·49-s − 1.37i·53-s + 1.30·59-s − 0.640·61-s + 0.366i·67-s + ⋯

Functional equation

Λ(s)=(7200s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(7200s/2 ΓC(s+1/2) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 72007200    =    25⋅32⋅522^{5} \cdot 3^{2} \cdot 5^{2}
Sign: 0.447−0.894i0.447 - 0.894i
Analytic conductor: 57.492257.4922
Root analytic conductor: 7.582367.58236
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ7200(6049,⋅)\chi_{7200} (6049, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 7200, ( :1/2), 0.447−0.894i)(2,\ 7200,\ (\ :1/2),\ 0.447 - 0.894i)

Particular Values

L(1)L(1) ≈\approx 2.1318479032.131847903
L(12)L(\frac12) ≈\approx 2.1318479032.131847903
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 1−3iT−7T2 1 - 3iT - 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1−5iT−13T2 1 - 5iT - 13T^{2}
17 1−17T2 1 - 17T^{2}
19 1−5T+19T2 1 - 5T + 19T^{2}
23 1+4iT−23T2 1 + 4iT - 23T^{2}
29 1−4T+29T2 1 - 4T + 29T^{2}
31 1−5T+31T2 1 - 5T + 31T^{2}
37 1+10iT−37T2 1 + 10iT - 37T^{2}
41 1−10T+41T2 1 - 10T + 41T^{2}
43 1−iT−43T2 1 - iT - 43T^{2}
47 1+2iT−47T2 1 + 2iT - 47T^{2}
53 1+10iT−53T2 1 + 10iT - 53T^{2}
59 1−10T+59T2 1 - 10T + 59T^{2}
61 1+5T+61T2 1 + 5T + 61T^{2}
67 1−3iT−67T2 1 - 3iT - 67T^{2}
71 1+10T+71T2 1 + 10T + 71T^{2}
73 1−10iT−73T2 1 - 10iT - 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1−14iT−83T2 1 - 14iT - 83T^{2}
89 1−16T+89T2 1 - 16T + 89T^{2}
97 1+5iT−97T2 1 + 5iT - 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

−8.148468579894745603695875477969, −7.28023168690919572646069183866, −6.63541625678875215429004032080, −5.94568846680347586150238349800, −5.27508510328405042018960804052, −4.50723559878007910352613320723, −3.75262958844375785241169807452, −2.64691516672709656661156680712, −2.17228386581809727127179070735, −0.942848077601048138011088464609, 0.67185663936785838554514551888, 1.34120108166275898542197366711, 2.83177302732851896102600997987, 3.28585768730193686191191002202, 4.25532822294967279801780018616, 4.90875745841107105449303967299, 5.72872715915044863802487067710, 6.37927225341814861272845710688, 7.36774563390800496799461939241, 7.63699792013487946406843668830

Graph of the ZZ-function along the critical line