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Properties

Label 4-1350e2-1.1-c1e2-0-18
Degree 44
Conductor 18225001822500
Sign 11
Analytic cond. 116.204116.204
Root an. cond. 3.283263.28326
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s + 4·8-s + 5·16-s + 8·17-s + 12·19-s − 4·23-s + 14·31-s + 6·32-s + 16·34-s + 24·38-s − 8·46-s + 4·47-s + 5·49-s − 6·53-s − 8·61-s + 28·62-s + 7·64-s + 24·68-s + 36·76-s − 10·83-s − 12·92-s + 8·94-s + 10·98-s − 12·106-s + 30·107-s + 20·109-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s + 5/4·16-s + 1.94·17-s + 2.75·19-s − 0.834·23-s + 2.51·31-s + 1.06·32-s + 2.74·34-s + 3.89·38-s − 1.17·46-s + 0.583·47-s + 5/7·49-s − 0.824·53-s − 1.02·61-s + 3.55·62-s + 7/8·64-s + 2.91·68-s + 4.12·76-s − 1.09·83-s − 1.25·92-s + 0.825·94-s + 1.01·98-s − 1.16·106-s + 2.90·107-s + 1.91·109-s + ⋯

Functional equation

Λ(s)=(1822500s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1822500s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 18225001822500    =    22⋅36⋅542^{2} \cdot 3^{6} \cdot 5^{4}
Sign: 11
Analytic conductor: 116.204116.204
Root analytic conductor: 3.283263.28326
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1822500, ( :1/2,1/2), 1)(4,\ 1822500,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 7.5834173747.583417374
L(12)L(\frac12) ≈\approx 7.5834173747.583417374
L(32)L(\frac{3}{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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C1C_1 (1−T)2 ( 1 - T )^{2}
3 1 1
5 1 1
good7C22C_2^2 1−5T2+p2T4 1 - 5 T^{2} + p^{2} T^{4}
11C22C_2^2 1+3T2+p2T4 1 + 3 T^{2} + p^{2} T^{4}
13C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
17C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
19C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
23C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
29C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
31C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}
37C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
41C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
43C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
47C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
53C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
59C22C_2^2 1+42T2+p2T4 1 + 42 T^{2} + p^{2} T^{4}
61C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
67C22C_2^2 1+58T2+p2T4 1 + 58 T^{2} + p^{2} T^{4}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C22C_2^2 1+127T2+p2T4 1 + 127 T^{2} + p^{2} T^{4}
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C2C_2 (1+5T+pT2)2 ( 1 + 5 T + p T^{2} )^{2}
89C22C_2^2 1+102T2+p2T4 1 + 102 T^{2} + p^{2} T^{4}
97C22C_2^2 1+175T2+p2T4 1 + 175 T^{2} + p^{2} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.831766250536615909555991793629, −9.745710883923792087820376688981, −8.999396774747180827227301432003, −8.521640902459070146262123743588, −7.80286480238678608669437815722, −7.76977121011768601913358247835, −7.41152490360258934930589333356, −6.92491019420454483770102890154, −6.23668807634405041476143501819, −6.03490131305999332310631451392, −5.60978920086988969726692862806, −5.12800961747859868254216974970, −4.85036360342555564090829170734, −4.31693309753682224821546583327, −3.53307271168479685023107989173, −3.48883838002187822386271183003, −2.80562676525849665288697615059, −2.48726936281705293448809254923, −1.31472464666321829271225485775, −1.09564415871161753804012423175, 1.09564415871161753804012423175, 1.31472464666321829271225485775, 2.48726936281705293448809254923, 2.80562676525849665288697615059, 3.48883838002187822386271183003, 3.53307271168479685023107989173, 4.31693309753682224821546583327, 4.85036360342555564090829170734, 5.12800961747859868254216974970, 5.60978920086988969726692862806, 6.03490131305999332310631451392, 6.23668807634405041476143501819, 6.92491019420454483770102890154, 7.41152490360258934930589333356, 7.76977121011768601913358247835, 7.80286480238678608669437815722, 8.521640902459070146262123743588, 8.999396774747180827227301432003, 9.745710883923792087820376688981, 9.831766250536615909555991793629

Graph of the ZZ-function along the critical line