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Properties

Label 2-200-1.1-c7-0-16
Degree 22
Conductor 200200
Sign 11
Analytic cond. 62.477062.4770
Root an. cond. 7.904237.90423
Motivic weight 77
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 69·3-s − 174·7-s + 2.57e3·9-s + 7.11e3·11-s + 468·13-s + 9.55e3·17-s − 4.26e4·19-s − 1.20e4·21-s + 7.75e4·23-s + 2.67e4·27-s − 6.13e4·29-s + 2.51e5·31-s + 4.90e5·33-s + 8.34e4·37-s + 3.22e4·39-s + 3.63e5·41-s + 3.41e4·43-s + 7.08e5·47-s − 7.93e5·49-s + 6.59e5·51-s + 8.91e5·53-s − 2.93e6·57-s + 2.80e6·59-s − 3.21e6·61-s − 4.47e5·63-s − 1.37e6·67-s + 5.34e6·69-s + ⋯
L(s)  = 1  + 1.47·3-s − 0.191·7-s + 1.17·9-s + 1.61·11-s + 0.0590·13-s + 0.471·17-s − 1.42·19-s − 0.282·21-s + 1.32·23-s + 0.261·27-s − 0.466·29-s + 1.51·31-s + 2.37·33-s + 0.270·37-s + 0.0871·39-s + 0.823·41-s + 0.0655·43-s + 0.995·47-s − 0.963·49-s + 0.695·51-s + 0.822·53-s − 2.10·57-s + 1.78·59-s − 1.81·61-s − 0.225·63-s − 0.557·67-s + 1.96·69-s + ⋯

Functional equation

Λ(s)=(200s/2 ΓC(s) L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
Λ(s)=(200s/2 ΓC(s+7/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 200200    =    23⋅522^{3} \cdot 5^{2}
Sign: 11
Analytic conductor: 62.477062.4770
Root analytic conductor: 7.904237.90423
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 200, ( :7/2), 1)(2,\ 200,\ (\ :7/2),\ 1)

Particular Values

L(4)L(4) ≈\approx 4.1471682074.147168207
L(12)L(\frac12) ≈\approx 4.1471682074.147168207
L(92)L(\frac{9}{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
5 1 1
good3 1−23pT+p7T2 1 - 23 p T + p^{7} T^{2}
7 1+174T+p7T2 1 + 174 T + p^{7} T^{2}
11 1−7111T+p7T2 1 - 7111 T + p^{7} T^{2}
13 1−36pT+p7T2 1 - 36 p T + p^{7} T^{2}
17 1−9555T+p7T2 1 - 9555 T + p^{7} T^{2}
19 1+42601T+p7T2 1 + 42601 T + p^{7} T^{2}
23 1−77526T+p7T2 1 - 77526 T + p^{7} T^{2}
29 1+61312T+p7T2 1 + 61312 T + p^{7} T^{2}
31 1−251710T+p7T2 1 - 251710 T + p^{7} T^{2}
37 1−83462T+p7T2 1 - 83462 T + p^{7} T^{2}
41 1−363477T+p7T2 1 - 363477 T + p^{7} T^{2}
43 1−34188T+p7T2 1 - 34188 T + p^{7} T^{2}
47 1−708812T+p7T2 1 - 708812 T + p^{7} T^{2}
53 1−891762T+p7T2 1 - 891762 T + p^{7} T^{2}
59 1−2809152T+p7T2 1 - 2809152 T + p^{7} T^{2}
61 1+3211510T+p7T2 1 + 3211510 T + p^{7} T^{2}
67 1+1372033T+p7T2 1 + 1372033 T + p^{7} T^{2}
71 1−4508308T+p7T2 1 - 4508308 T + p^{7} T^{2}
73 1+628179T+p7T2 1 + 628179 T + p^{7} T^{2}
79 1−6130474T+p7T2 1 - 6130474 T + p^{7} T^{2}
83 1+9921981T+p7T2 1 + 9921981 T + p^{7} T^{2}
89 1−1806599T+p7T2 1 - 1806599 T + p^{7} T^{2}
97 1+11676482T+p7T2 1 + 11676482 T + p^{7} T^{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

−11.14871729964829133129510440248, −9.866596046456988818755986768251, −9.039990701363541108747315157096, −8.425822955599581930428040179437, −7.23483959844173527970931189534, −6.22769979597744438789903169300, −4.39450799148665595811187862923, −3.48331598556110100812024700039, −2.36269973482364577566660554879, −1.09316959064928042871523792270, 1.09316959064928042871523792270, 2.36269973482364577566660554879, 3.48331598556110100812024700039, 4.39450799148665595811187862923, 6.22769979597744438789903169300, 7.23483959844173527970931189534, 8.425822955599581930428040179437, 9.039990701363541108747315157096, 9.866596046456988818755986768251, 11.14871729964829133129510440248

Graph of the ZZ-function along the critical line