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Properties

Label 4-112e2-1.1-c1e2-0-2
Degree 44
Conductor 1254412544
Sign 11
Analytic cond. 0.7998160.799816
Root an. cond. 0.9456870.945687
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s + 4·5-s − 8·10-s + 2·11-s − 4·16-s − 4·17-s + 4·19-s + 8·20-s − 4·22-s + 8·25-s + 14·29-s − 16·31-s + 8·32-s + 8·34-s − 10·37-s − 8·38-s − 2·43-s + 4·44-s − 24·47-s − 49-s − 16·50-s − 2·53-s + 8·55-s − 28·58-s + 16·59-s + 12·61-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 1.78·5-s − 2.52·10-s + 0.603·11-s − 16-s − 0.970·17-s + 0.917·19-s + 1.78·20-s − 0.852·22-s + 8/5·25-s + 2.59·29-s − 2.87·31-s + 1.41·32-s + 1.37·34-s − 1.64·37-s − 1.29·38-s − 0.304·43-s + 0.603·44-s − 3.50·47-s − 1/7·49-s − 2.26·50-s − 0.274·53-s + 1.07·55-s − 3.67·58-s + 2.08·59-s + 1.53·61-s + ⋯

Functional equation

Λ(s)=(12544s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(12544s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1254412544    =    28⋅722^{8} \cdot 7^{2}
Sign: 11
Analytic conductor: 0.7998160.799816
Root analytic conductor: 0.9456870.945687
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 12544, ( :1/2,1/2), 1)(4,\ 12544,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.71144888720.7114488872
L(12)L(\frac12) ≈\approx 0.71144888720.7114488872
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)
bad2C2C_2 1+pT+pT2 1 + p T + p T^{2}
7C2C_2 1+T2 1 + T^{2}
good3C22C_2^2 1+p2T4 1 + p^{2} T^{4}
5C22C_2^2 1−4T+8T2−4pT3+p2T4 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4}
11C22C_2^2 1−2T+2T2−2pT3+p2T4 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4}
13C22C_2^2 1+p2T4 1 + p^{2} T^{4}
17C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
19C22C_2^2 1−4T+8T2−4pT3+p2T4 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4}
23C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
29C2C_2 (1−10T+pT2)(1−4T+pT2) ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} )
31C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
37C2C_2 (1−2T+pT2)(1+12T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} )
41C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
43C22C_2^2 1+2T+2T2+2pT3+p2T4 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
47C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
53C22C_2^2 1+2T+2T2+2pT3+p2T4 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
59C22C_2^2 1−16T+128T2−16pT3+p2T4 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4}
61C22C_2^2 1−12T+72T2−12pT3+p2T4 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4}
67C22C_2^2 1−6T+18T2−6pT3+p2T4 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4}
71C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
73C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
79C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
83C22C_2^2 1+20T+200T2+20pT3+p2T4 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4}
89C22C_2^2 1+18T2+p2T4 1 + 18 T^{2} + p^{2} T^{4}
97C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
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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

−14.07709379257313501434929525229, −13.21080224703493065311584299987, −13.10435677069817317715597291629, −12.33374275476837685879156416076, −11.41893359581473435188152530904, −11.26811707037460048297898716732, −10.42474346384492172731049067434, −10.00937462620166698547838782373, −9.683924366003053898879873910454, −9.197203073722881335501337628871, −8.540009028955318294445626349058, −8.332459713466063053230675288684, −7.11194844754978421779119417990, −6.84081957360369013901511394052, −6.27061469703395830923553716435, −5.33221791626940470964482586284, −4.84926899829197202780201987783, −3.49326979522568431710341003998, −2.23309122526353656923885425256, −1.51802328772081144901566458296, 1.51802328772081144901566458296, 2.23309122526353656923885425256, 3.49326979522568431710341003998, 4.84926899829197202780201987783, 5.33221791626940470964482586284, 6.27061469703395830923553716435, 6.84081957360369013901511394052, 7.11194844754978421779119417990, 8.332459713466063053230675288684, 8.540009028955318294445626349058, 9.197203073722881335501337628871, 9.683924366003053898879873910454, 10.00937462620166698547838782373, 10.42474346384492172731049067434, 11.26811707037460048297898716732, 11.41893359581473435188152530904, 12.33374275476837685879156416076, 13.10435677069817317715597291629, 13.21080224703493065311584299987, 14.07709379257313501434929525229

Graph of the ZZ-function along the critical line