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Properties

Label 2-2205-1.1-c3-0-118
Degree 22
Conductor 22052205
Sign 11
Analytic cond. 130.099130.099
Root an. cond. 11.406111.4061
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.91·2-s + 16.1·4-s + 5·5-s + 40.1·8-s + 24.5·10-s − 11.2·11-s − 31.5·13-s + 68.0·16-s + 67.9·17-s + 158.·19-s + 80.8·20-s − 55.0·22-s − 41.6·23-s + 25·25-s − 155.·26-s + 278.·29-s + 35.4·31-s + 13.3·32-s + 334.·34-s − 35.5·37-s + 778.·38-s + 200.·40-s − 154.·41-s − 309.·43-s − 181.·44-s − 204.·46-s + 277.·47-s + ⋯
L(s)  = 1  + 1.73·2-s + 2.02·4-s + 0.447·5-s + 1.77·8-s + 0.777·10-s − 0.307·11-s − 0.673·13-s + 1.06·16-s + 0.970·17-s + 1.91·19-s + 0.903·20-s − 0.533·22-s − 0.378·23-s + 0.200·25-s − 1.17·26-s + 1.78·29-s + 0.205·31-s + 0.0735·32-s + 1.68·34-s − 0.157·37-s + 3.32·38-s + 0.793·40-s − 0.588·41-s − 1.09·43-s − 0.620·44-s − 0.657·46-s + 0.859·47-s + ⋯

Functional equation

Λ(s)=(2205s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(2205s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 22052205    =    32⋅5⋅723^{2} \cdot 5 \cdot 7^{2}
Sign: 11
Analytic conductor: 130.099130.099
Root analytic conductor: 11.406111.4061
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2205, ( :3/2), 1)(2,\ 2205,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 8.1501938998.150193899
L(12)L(\frac12) ≈\approx 8.1501938998.150193899
L(52)L(\frac{5}{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)
bad3 1 1
5 1−5T 1 - 5T
7 1 1
good2 1−4.91T+8T2 1 - 4.91T + 8T^{2}
11 1+11.2T+1.33e3T2 1 + 11.2T + 1.33e3T^{2}
13 1+31.5T+2.19e3T2 1 + 31.5T + 2.19e3T^{2}
17 1−67.9T+4.91e3T2 1 - 67.9T + 4.91e3T^{2}
19 1−158.T+6.85e3T2 1 - 158.T + 6.85e3T^{2}
23 1+41.6T+1.21e4T2 1 + 41.6T + 1.21e4T^{2}
29 1−278.T+2.43e4T2 1 - 278.T + 2.43e4T^{2}
31 1−35.4T+2.97e4T2 1 - 35.4T + 2.97e4T^{2}
37 1+35.5T+5.06e4T2 1 + 35.5T + 5.06e4T^{2}
41 1+154.T+6.89e4T2 1 + 154.T + 6.89e4T^{2}
43 1+309.T+7.95e4T2 1 + 309.T + 7.95e4T^{2}
47 1−277.T+1.03e5T2 1 - 277.T + 1.03e5T^{2}
53 1+79.8T+1.48e5T2 1 + 79.8T + 1.48e5T^{2}
59 1−901.T+2.05e5T2 1 - 901.T + 2.05e5T^{2}
61 1−514.T+2.26e5T2 1 - 514.T + 2.26e5T^{2}
67 1−774.T+3.00e5T2 1 - 774.T + 3.00e5T^{2}
71 1+697.T+3.57e5T2 1 + 697.T + 3.57e5T^{2}
73 1−441.T+3.89e5T2 1 - 441.T + 3.89e5T^{2}
79 1+305.T+4.93e5T2 1 + 305.T + 4.93e5T^{2}
83 1−925.T+5.71e5T2 1 - 925.T + 5.71e5T^{2}
89 1+46.7T+7.04e5T2 1 + 46.7T + 7.04e5T^{2}
97 1−779.T+9.12e5T2 1 - 779.T + 9.12e5T^{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

−8.545007667596842795443620804789, −7.57241017584103997469838721731, −6.91795077661738455373510504233, −6.09773922532841440499698581917, −5.19108653255088530237515353122, −5.02688896572114882219457928835, −3.78140585033466480384801039899, −3.04451295539954882201981565478, −2.28986089814839037503177917420, −1.02797862745646075823150411519, 1.02797862745646075823150411519, 2.28986089814839037503177917420, 3.04451295539954882201981565478, 3.78140585033466480384801039899, 5.02688896572114882219457928835, 5.19108653255088530237515353122, 6.09773922532841440499698581917, 6.91795077661738455373510504233, 7.57241017584103997469838721731, 8.545007667596842795443620804789

Graph of the ZZ-function along the critical line