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Properties

Label 2-2205-1.1-c3-0-34
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  − 4.70·2-s + 14.1·4-s − 5·5-s − 28.7·8-s + 23.5·10-s − 24.5·11-s + 35.0·13-s + 22.1·16-s − 18.4·17-s + 67.4·19-s − 70.5·20-s + 115.·22-s + 145.·23-s + 25·25-s − 164.·26-s − 214.·29-s + 88.6·31-s + 125.·32-s + 86.5·34-s + 162.·37-s − 316.·38-s + 143.·40-s − 337.·41-s + 122.·43-s − 346.·44-s − 684.·46-s + 354.·47-s + ⋯
L(s)  = 1  − 1.66·2-s + 1.76·4-s − 0.447·5-s − 1.26·8-s + 0.743·10-s − 0.674·11-s + 0.747·13-s + 0.345·16-s − 0.262·17-s + 0.813·19-s − 0.788·20-s + 1.12·22-s + 1.32·23-s + 0.200·25-s − 1.24·26-s − 1.37·29-s + 0.513·31-s + 0.694·32-s + 0.436·34-s + 0.720·37-s − 1.35·38-s + 0.567·40-s − 1.28·41-s + 0.433·43-s − 1.18·44-s − 2.19·46-s + 1.09·47-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 0.72031540800.7203154080
L(12)L(\frac12) ≈\approx 0.72031540800.7203154080
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+4.70T+8T2 1 + 4.70T + 8T^{2}
11 1+24.5T+1.33e3T2 1 + 24.5T + 1.33e3T^{2}
13 1−35.0T+2.19e3T2 1 - 35.0T + 2.19e3T^{2}
17 1+18.4T+4.91e3T2 1 + 18.4T + 4.91e3T^{2}
19 1−67.4T+6.85e3T2 1 - 67.4T + 6.85e3T^{2}
23 1−145.T+1.21e4T2 1 - 145.T + 1.21e4T^{2}
29 1+214.T+2.43e4T2 1 + 214.T + 2.43e4T^{2}
31 1−88.6T+2.97e4T2 1 - 88.6T + 2.97e4T^{2}
37 1−162.T+5.06e4T2 1 - 162.T + 5.06e4T^{2}
41 1+337.T+6.89e4T2 1 + 337.T + 6.89e4T^{2}
43 1−122.T+7.95e4T2 1 - 122.T + 7.95e4T^{2}
47 1−354.T+1.03e5T2 1 - 354.T + 1.03e5T^{2}
53 1+676.T+1.48e5T2 1 + 676.T + 1.48e5T^{2}
59 1−501.T+2.05e5T2 1 - 501.T + 2.05e5T^{2}
61 1−708.T+2.26e5T2 1 - 708.T + 2.26e5T^{2}
67 1+907.T+3.00e5T2 1 + 907.T + 3.00e5T^{2}
71 1+430.T+3.57e5T2 1 + 430.T + 3.57e5T^{2}
73 1+41.3T+3.89e5T2 1 + 41.3T + 3.89e5T^{2}
79 1−890.T+4.93e5T2 1 - 890.T + 4.93e5T^{2}
83 1+1.05e3T+5.71e5T2 1 + 1.05e3T + 5.71e5T^{2}
89 1−1.47e3T+7.04e5T2 1 - 1.47e3T + 7.04e5T^{2}
97 1+555.T+9.12e5T2 1 + 555.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

−8.765023127653518868449475180324, −8.011307591654896696318463065031, −7.43039046950796777063564665707, −6.77500121526243522454268513281, −5.76951821820016139012997102054, −4.75093120025812212571984131523, −3.50589041345024107644173498813, −2.55841064261338600173888301568, −1.41669654627431476667269737316, −0.52963839797465127935562734935, 0.52963839797465127935562734935, 1.41669654627431476667269737316, 2.55841064261338600173888301568, 3.50589041345024107644173498813, 4.75093120025812212571984131523, 5.76951821820016139012997102054, 6.77500121526243522454268513281, 7.43039046950796777063564665707, 8.011307591654896696318463065031, 8.765023127653518868449475180324

Graph of the ZZ-function along the critical line