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Properties

Label 2-1200-1.1-c3-0-21
Degree 22
Conductor 12001200
Sign 11
Analytic cond. 70.802270.8022
Root an. cond. 8.414408.41440
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  + 3·3-s + 33.0·7-s + 9·9-s − 48.3·11-s − 60.3·13-s − 17.7·17-s + 130.·19-s + 99.2·21-s − 70.8·23-s + 27·27-s + 104.·29-s + 210.·31-s − 145.·33-s + 300.·37-s − 181.·39-s + 240.·41-s + 108·43-s + 278.·47-s + 750.·49-s − 53.3·51-s + 328.·53-s + 392.·57-s − 889.·59-s − 241.·61-s + 297.·63-s − 103.·67-s − 212.·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.78·7-s + 0.333·9-s − 1.32·11-s − 1.28·13-s − 0.253·17-s + 1.58·19-s + 1.03·21-s − 0.642·23-s + 0.192·27-s + 0.669·29-s + 1.21·31-s − 0.765·33-s + 1.33·37-s − 0.743·39-s + 0.914·41-s + 0.383·43-s + 0.865·47-s + 2.18·49-s − 0.146·51-s + 0.851·53-s + 0.912·57-s − 1.96·59-s − 0.506·61-s + 0.595·63-s − 0.189·67-s − 0.370·69-s + ⋯

Functional equation

Λ(s)=(1200s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1200s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12001200    =    24⋅3⋅522^{4} \cdot 3 \cdot 5^{2}
Sign: 11
Analytic conductor: 70.802270.8022
Root analytic conductor: 8.414408.41440
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1200, ( :3/2), 1)(2,\ 1200,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 3.1199905093.119990509
L(12)L(\frac12) ≈\approx 3.1199905093.119990509
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)
bad2 1 1
3 1−3T 1 - 3T
5 1 1
good7 1−33.0T+343T2 1 - 33.0T + 343T^{2}
11 1+48.3T+1.33e3T2 1 + 48.3T + 1.33e3T^{2}
13 1+60.3T+2.19e3T2 1 + 60.3T + 2.19e3T^{2}
17 1+17.7T+4.91e3T2 1 + 17.7T + 4.91e3T^{2}
19 1−130.T+6.85e3T2 1 - 130.T + 6.85e3T^{2}
23 1+70.8T+1.21e4T2 1 + 70.8T + 1.21e4T^{2}
29 1−104.T+2.43e4T2 1 - 104.T + 2.43e4T^{2}
31 1−210.T+2.97e4T2 1 - 210.T + 2.97e4T^{2}
37 1−300.T+5.06e4T2 1 - 300.T + 5.06e4T^{2}
41 1−240.T+6.89e4T2 1 - 240.T + 6.89e4T^{2}
43 1−108T+7.95e4T2 1 - 108T + 7.95e4T^{2}
47 1−278.T+1.03e5T2 1 - 278.T + 1.03e5T^{2}
53 1−328.T+1.48e5T2 1 - 328.T + 1.48e5T^{2}
59 1+889.T+2.05e5T2 1 + 889.T + 2.05e5T^{2}
61 1+241.T+2.26e5T2 1 + 241.T + 2.26e5T^{2}
67 1+103.T+3.00e5T2 1 + 103.T + 3.00e5T^{2}
71 1−277.T+3.57e5T2 1 - 277.T + 3.57e5T^{2}
73 1−274.T+3.89e5T2 1 - 274.T + 3.89e5T^{2}
79 1+366.T+4.93e5T2 1 + 366.T + 4.93e5T^{2}
83 1+57.7T+5.71e5T2 1 + 57.7T + 5.71e5T^{2}
89 1+203.T+7.04e5T2 1 + 203.T + 7.04e5T^{2}
97 1−1.28e3T+9.12e5T2 1 - 1.28e3T + 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

−9.372783420025994038417886062849, −8.340661919045081631091633034023, −7.68880235023874252580015271091, −7.42556424170616120194743333112, −5.82023630005094069514386690949, −4.90238083442971768066309319949, −4.44863420620650766132696366503, −2.84867373886782349622594371706, −2.19081336113120710667779492294, −0.902478242542447817355648283085, 0.902478242542447817355648283085, 2.19081336113120710667779492294, 2.84867373886782349622594371706, 4.44863420620650766132696366503, 4.90238083442971768066309319949, 5.82023630005094069514386690949, 7.42556424170616120194743333112, 7.68880235023874252580015271091, 8.340661919045081631091633034023, 9.372783420025994038417886062849

Graph of the ZZ-function along the critical line