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Properties

Label 2-7200-40.29-c1-0-66
Degree 22
Conductor 72007200
Sign −0.316+0.948i-0.316 + 0.948i
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  + 2i·7-s + 4i·11-s − 6i·17-s + 4i·19-s − 4i·23-s − 6i·29-s − 10·31-s + 4·37-s − 10·41-s − 4·43-s + 4i·47-s + 3·49-s − 10·53-s − 8i·59-s − 8i·61-s + ⋯
L(s)  = 1  + 0.755i·7-s + 1.20i·11-s − 1.45i·17-s + 0.917i·19-s − 0.834i·23-s − 1.11i·29-s − 1.79·31-s + 0.657·37-s − 1.56·41-s − 0.609·43-s + 0.583i·47-s + 0.428·49-s − 1.37·53-s − 1.04i·59-s − 1.02i·61-s + ⋯

Functional equation

Λ(s)=(7200s/2 ΓC(s) L(s)=((−0.316+0.948i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(7200s/2 ΓC(s+1/2) L(s)=((−0.316+0.948i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 72007200    =    25⋅32⋅522^{5} \cdot 3^{2} \cdot 5^{2}
Sign: −0.316+0.948i-0.316 + 0.948i
Analytic conductor: 57.492257.4922
Root analytic conductor: 7.582367.58236
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ7200(2449,⋅)\chi_{7200} (2449, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 7200, ( :1/2), −0.316+0.948i)(2,\ 7200,\ (\ :1/2),\ -0.316 + 0.948i)

Particular Values

L(1)L(1) ≈\approx 0.66799043370.6679904337
L(12)L(\frac12) ≈\approx 0.66799043370.6679904337
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 1−2iT−7T2 1 - 2iT - 7T^{2}
11 1−4iT−11T2 1 - 4iT - 11T^{2}
13 1+13T2 1 + 13T^{2}
17 1+6iT−17T2 1 + 6iT - 17T^{2}
19 1−4iT−19T2 1 - 4iT - 19T^{2}
23 1+4iT−23T2 1 + 4iT - 23T^{2}
29 1+6iT−29T2 1 + 6iT - 29T^{2}
31 1+10T+31T2 1 + 10T + 31T^{2}
37 1−4T+37T2 1 - 4T + 37T^{2}
41 1+10T+41T2 1 + 10T + 41T^{2}
43 1+4T+43T2 1 + 4T + 43T^{2}
47 1−4iT−47T2 1 - 4iT - 47T^{2}
53 1+10T+53T2 1 + 10T + 53T^{2}
59 1+8iT−59T2 1 + 8iT - 59T^{2}
61 1+8iT−61T2 1 + 8iT - 61T^{2}
67 1−12T+67T2 1 - 12T + 67T^{2}
71 1+4T+71T2 1 + 4T + 71T^{2}
73 1−10iT−73T2 1 - 10iT - 73T^{2}
79 1+14T+79T2 1 + 14T + 79T^{2}
83 1+83T2 1 + 83T^{2}
89 1−14T+89T2 1 - 14T + 89T^{2}
97 1−10iT−97T2 1 - 10iT - 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

−7.72245141839560348990795671873, −7.01524480659532236746056979592, −6.38453034787917308936478963289, −5.50049250889242634962875459496, −4.94693218148454056287421844696, −4.19761845725705193880022237990, −3.24108789999648633850882728161, −2.35321444672233488719118895608, −1.70213899085251737986528195714, −0.16654237439607242280517030264, 1.07037968262748505843695831373, 1.94932438853790802435305219186, 3.30975093088840062450932444575, 3.57631236054534981209029571192, 4.54387220811817044759358299572, 5.41136457983907222899504045179, 5.99489350071448969992445769950, 6.82572682119283573777787240605, 7.37275693547771654853789522853, 8.174724527848889956806089709828

Graph of the ZZ-function along the critical line