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Properties

Label 2-7200-5.4-c1-0-69
Degree 22
Conductor 72007200
Sign −0.447+0.894i-0.447 + 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  − 4.47i·7-s − 2.23·11-s + 4i·13-s − 7i·17-s + 6.70·19-s + 4.47i·23-s + 4.47·31-s − 2i·37-s − 5·41-s − 8.94i·47-s − 13.0·49-s − 6i·53-s + 8.94·59-s + 10·61-s + 2.23i·67-s + ⋯
L(s)  = 1  − 1.69i·7-s − 0.674·11-s + 1.10i·13-s − 1.69i·17-s + 1.53·19-s + 0.932i·23-s + 0.803·31-s − 0.328i·37-s − 0.780·41-s − 1.30i·47-s − 1.85·49-s − 0.824i·53-s + 1.16·59-s + 1.28·61-s + 0.273i·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+0.894i-0.447 + 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 1.5844532511.584453251
L(12)L(\frac12) ≈\approx 1.5844532511.584453251
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+4.47iT−7T2 1 + 4.47iT - 7T^{2}
11 1+2.23T+11T2 1 + 2.23T + 11T^{2}
13 1−4iT−13T2 1 - 4iT - 13T^{2}
17 1+7iT−17T2 1 + 7iT - 17T^{2}
19 1−6.70T+19T2 1 - 6.70T + 19T^{2}
23 1−4.47iT−23T2 1 - 4.47iT - 23T^{2}
29 1+29T2 1 + 29T^{2}
31 1−4.47T+31T2 1 - 4.47T + 31T^{2}
37 1+2iT−37T2 1 + 2iT - 37T^{2}
41 1+5T+41T2 1 + 5T + 41T^{2}
43 1−43T2 1 - 43T^{2}
47 1+8.94iT−47T2 1 + 8.94iT - 47T^{2}
53 1+6iT−53T2 1 + 6iT - 53T^{2}
59 1−8.94T+59T2 1 - 8.94T + 59T^{2}
61 1−10T+61T2 1 - 10T + 61T^{2}
67 1−2.23iT−67T2 1 - 2.23iT - 67T^{2}
71 1−8.94T+71T2 1 - 8.94T + 71T^{2}
73 1+9iT−73T2 1 + 9iT - 73T^{2}
79 1−4.47T+79T2 1 - 4.47T + 79T^{2}
83 1−11.1iT−83T2 1 - 11.1iT - 83T^{2}
89 1+5T+89T2 1 + 5T + 89T^{2}
97 1+2iT−97T2 1 + 2iT - 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

−7.51632126960610043415229032097, −7.07285863866384932498644139999, −6.62989510940219012964138335644, −5.26463914749291873568210757351, −5.01829961303281057425261312630, −3.99259519659740165898230981111, −3.44713520898222059297840299683, −2.46638329762792549391810432918, −1.29348053404205404734656024124, −0.43061105971833427041287858923, 1.12233395391731144027394747420, 2.30861678740120204224271111606, 2.85720539381368682477026792647, 3.65561792616361294338861589601, 4.83719862463972772640668932897, 5.42203298465228833083737670731, 5.92542995203487407829889541354, 6.58268991199515317411171588345, 7.68329358886812684893898113016, 8.253529210944521344842222668456

Graph of the ZZ-function along the critical line