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Properties

Label 2-2205-1.1-c3-0-10
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  − 5.20·2-s + 19.0·4-s − 5·5-s − 57.6·8-s + 26.0·10-s − 56.6·11-s + 43.4·13-s + 147.·16-s − 39.8·17-s − 52.3·19-s − 95.3·20-s + 294.·22-s + 53.5·23-s + 25·25-s − 225.·26-s − 49.6·29-s + 73.7·31-s − 305.·32-s + 207.·34-s − 307.·37-s + 272.·38-s + 288.·40-s + 292.·41-s − 365.·43-s − 1.08e3·44-s − 278.·46-s − 442.·47-s + ⋯
L(s)  = 1  − 1.83·2-s + 2.38·4-s − 0.447·5-s − 2.54·8-s + 0.822·10-s − 1.55·11-s + 0.926·13-s + 2.30·16-s − 0.568·17-s − 0.632·19-s − 1.06·20-s + 2.85·22-s + 0.485·23-s + 0.200·25-s − 1.70·26-s − 0.317·29-s + 0.427·31-s − 1.68·32-s + 1.04·34-s − 1.36·37-s + 1.16·38-s + 1.13·40-s + 1.11·41-s − 1.29·43-s − 3.70·44-s − 0.892·46-s − 1.37·47-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 0.31848906110.3184890611
L(12)L(\frac12) ≈\approx 0.31848906110.3184890611
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+5.20T+8T2 1 + 5.20T + 8T^{2}
11 1+56.6T+1.33e3T2 1 + 56.6T + 1.33e3T^{2}
13 1−43.4T+2.19e3T2 1 - 43.4T + 2.19e3T^{2}
17 1+39.8T+4.91e3T2 1 + 39.8T + 4.91e3T^{2}
19 1+52.3T+6.85e3T2 1 + 52.3T + 6.85e3T^{2}
23 1−53.5T+1.21e4T2 1 - 53.5T + 1.21e4T^{2}
29 1+49.6T+2.43e4T2 1 + 49.6T + 2.43e4T^{2}
31 1−73.7T+2.97e4T2 1 - 73.7T + 2.97e4T^{2}
37 1+307.T+5.06e4T2 1 + 307.T + 5.06e4T^{2}
41 1−292.T+6.89e4T2 1 - 292.T + 6.89e4T^{2}
43 1+365.T+7.95e4T2 1 + 365.T + 7.95e4T^{2}
47 1+442.T+1.03e5T2 1 + 442.T + 1.03e5T^{2}
53 1+25.7T+1.48e5T2 1 + 25.7T + 1.48e5T^{2}
59 1−376.T+2.05e5T2 1 - 376.T + 2.05e5T^{2}
61 1+632.T+2.26e5T2 1 + 632.T + 2.26e5T^{2}
67 1−511.T+3.00e5T2 1 - 511.T + 3.00e5T^{2}
71 1+134.T+3.57e5T2 1 + 134.T + 3.57e5T^{2}
73 1+409.T+3.89e5T2 1 + 409.T + 3.89e5T^{2}
79 1+926.T+4.93e5T2 1 + 926.T + 4.93e5T^{2}
83 1−296.T+5.71e5T2 1 - 296.T + 5.71e5T^{2}
89 1−488.T+7.04e5T2 1 - 488.T + 7.04e5T^{2}
97 1−475.T+9.12e5T2 1 - 475.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

−8.444648816804266169228558962011, −8.295984246461963348823047958578, −7.39586293797013354895189064437, −6.74260874749952116528547544882, −5.88393035501710146999830574013, −4.78121388250290639271504972806, −3.39608942089151859182867711145, −2.49486395269559300339835338332, −1.53246253478909874278807853100, −0.33759025064208613497695651845, 0.33759025064208613497695651845, 1.53246253478909874278807853100, 2.49486395269559300339835338332, 3.39608942089151859182867711145, 4.78121388250290639271504972806, 5.88393035501710146999830574013, 6.74260874749952116528547544882, 7.39586293797013354895189064437, 8.295984246461963348823047958578, 8.444648816804266169228558962011

Graph of the ZZ-function along the critical line