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Properties

Label 2-7200-1.1-c1-0-38
Degree 22
Conductor 72007200
Sign 11
Analytic cond. 57.492257.4922
Root an. cond. 7.582367.58236
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.23·7-s − 7.70·23-s + 6·29-s − 4.47·41-s + 6.76·43-s − 0.291·47-s + 20.4·49-s + 13.4·61-s + 14.1·67-s + 4.29·83-s − 6·89-s + 18·101-s − 2.18·103-s + 19.7·107-s − 13.4·109-s + ⋯
L(s)  = 1  + 1.97·7-s − 1.60·23-s + 1.11·29-s − 0.698·41-s + 1.03·43-s − 0.0425·47-s + 2.91·49-s + 1.71·61-s + 1.73·67-s + 0.471·83-s − 0.635·89-s + 1.79·101-s − 0.214·103-s + 1.90·107-s − 1.28·109-s + ⋯

Functional equation

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

Invariants

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

Particular Values

L(1)L(1) ≈\approx 2.7067105802.706710580
L(12)L(\frac12) ≈\approx 2.7067105802.706710580
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 1−5.23T+7T2 1 - 5.23T + 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1+13T2 1 + 13T^{2}
17 1+17T2 1 + 17T^{2}
19 1+19T2 1 + 19T^{2}
23 1+7.70T+23T2 1 + 7.70T + 23T^{2}
29 1−6T+29T2 1 - 6T + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1+37T2 1 + 37T^{2}
41 1+4.47T+41T2 1 + 4.47T + 41T^{2}
43 1−6.76T+43T2 1 - 6.76T + 43T^{2}
47 1+0.291T+47T2 1 + 0.291T + 47T^{2}
53 1+53T2 1 + 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1−13.4T+61T2 1 - 13.4T + 61T^{2}
67 1−14.1T+67T2 1 - 14.1T + 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1+73T2 1 + 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1−4.29T+83T2 1 - 4.29T + 83T^{2}
89 1+6T+89T2 1 + 6T + 89T^{2}
97 1+97T2 1 + 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

−8.096559439968840208930334390025, −7.38196287531328426459859164677, −6.55304506873024720074428570896, −5.67834446320787553307869061323, −5.07503917300434549897758671992, −4.40062423072192247189511174106, −3.76033628270095452825522072513, −2.46129744287166072400837008542, −1.84434307512047331684423106027, −0.869168304842140774857568928108, 0.869168304842140774857568928108, 1.84434307512047331684423106027, 2.46129744287166072400837008542, 3.76033628270095452825522072513, 4.40062423072192247189511174106, 5.07503917300434549897758671992, 5.67834446320787553307869061323, 6.55304506873024720074428570896, 7.38196287531328426459859164677, 8.096559439968840208930334390025

Graph of the ZZ-function along the critical line