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Properties

Label 2-2205-1.1-c3-0-14
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  + 1.70·2-s − 5.10·4-s − 5·5-s − 22.2·8-s − 8.50·10-s − 37.4·11-s − 29.0·13-s + 2.89·16-s + 58.4·17-s + 54.5·19-s + 25.5·20-s − 63.6·22-s − 161.·23-s + 25·25-s − 49.3·26-s − 137.·29-s − 154.·31-s + 183.·32-s + 99.4·34-s − 350.·37-s + 92.8·38-s + 111.·40-s + 353.·41-s − 518.·43-s + 190.·44-s − 275.·46-s − 542.·47-s + ⋯
L(s)  = 1  + 0.601·2-s − 0.638·4-s − 0.447·5-s − 0.985·8-s − 0.269·10-s − 1.02·11-s − 0.619·13-s + 0.0452·16-s + 0.833·17-s + 0.659·19-s + 0.285·20-s − 0.616·22-s − 1.46·23-s + 0.200·25-s − 0.372·26-s − 0.880·29-s − 0.896·31-s + 1.01·32-s + 0.501·34-s − 1.55·37-s + 0.396·38-s + 0.440·40-s + 1.34·41-s − 1.83·43-s + 0.654·44-s − 0.881·46-s − 1.68·47-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 0.81939639600.8193963960
L(12)L(\frac12) ≈\approx 0.81939639600.8193963960
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 1−1.70T+8T2 1 - 1.70T + 8T^{2}
11 1+37.4T+1.33e3T2 1 + 37.4T + 1.33e3T^{2}
13 1+29.0T+2.19e3T2 1 + 29.0T + 2.19e3T^{2}
17 1−58.4T+4.91e3T2 1 - 58.4T + 4.91e3T^{2}
19 1−54.5T+6.85e3T2 1 - 54.5T + 6.85e3T^{2}
23 1+161.T+1.21e4T2 1 + 161.T + 1.21e4T^{2}
29 1+137.T+2.43e4T2 1 + 137.T + 2.43e4T^{2}
31 1+154.T+2.97e4T2 1 + 154.T + 2.97e4T^{2}
37 1+350.T+5.06e4T2 1 + 350.T + 5.06e4T^{2}
41 1−353.T+6.89e4T2 1 - 353.T + 6.89e4T^{2}
43 1+518.T+7.95e4T2 1 + 518.T + 7.95e4T^{2}
47 1+542.T+1.03e5T2 1 + 542.T + 1.03e5T^{2}
53 1+305.T+1.48e5T2 1 + 305.T + 1.48e5T^{2}
59 1−14.6T+2.05e5T2 1 - 14.6T + 2.05e5T^{2}
61 1−171.T+2.26e5T2 1 - 171.T + 2.26e5T^{2}
67 1−551.T+3.00e5T2 1 - 551.T + 3.00e5T^{2}
71 1−120.T+3.57e5T2 1 - 120.T + 3.57e5T^{2}
73 1+284.T+3.89e5T2 1 + 284.T + 3.89e5T^{2}
79 1−941.T+4.93e5T2 1 - 941.T + 4.93e5T^{2}
83 1−377.T+5.71e5T2 1 - 377.T + 5.71e5T^{2}
89 1+677.T+7.04e5T2 1 + 677.T + 7.04e5T^{2}
97 1−1.22e3T+9.12e5T2 1 - 1.22e3T + 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

−8.552390438446042517550388443818, −7.928121915751284233212196480082, −7.28162703920366389412844994007, −6.10789216281293128148667186853, −5.29085316505930329359099282207, −4.85458081491807854820364249910, −3.71303461658186779984528906303, −3.21303841623059107067077121610, −1.94340258328469866153911550303, −0.35838979906258974904439614630, 0.35838979906258974904439614630, 1.94340258328469866153911550303, 3.21303841623059107067077121610, 3.71303461658186779984528906303, 4.85458081491807854820364249910, 5.29085316505930329359099282207, 6.10789216281293128148667186853, 7.28162703920366389412844994007, 7.928121915751284233212196480082, 8.552390438446042517550388443818

Graph of the ZZ-function along the critical line