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Properties

Label 4-312e2-1.1-c1e2-0-31
Degree 44
Conductor 9734497344
Sign 11
Analytic cond. 6.206736.20673
Root an. cond. 1.578391.57839
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 + 2·3-s + 2·4-s + 4·5-s − 4·6-s + 6·7-s + 3·9-s − 8·10-s − 8·11-s + 4·12-s + 4·13-s − 12·14-s + 8·15-s − 4·16-s − 6·18-s − 6·19-s + 8·20-s + 12·21-s + 16·22-s + 8·23-s + 8·25-s − 8·26-s + 4·27-s + 12·28-s − 16·30-s + 2·31-s + 8·32-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 4-s + 1.78·5-s − 1.63·6-s + 2.26·7-s + 9-s − 2.52·10-s − 2.41·11-s + 1.15·12-s + 1.10·13-s − 3.20·14-s + 2.06·15-s − 16-s − 1.41·18-s − 1.37·19-s + 1.78·20-s + 2.61·21-s + 3.41·22-s + 1.66·23-s + 8/5·25-s − 1.56·26-s + 0.769·27-s + 2.26·28-s − 2.92·30-s + 0.359·31-s + 1.41·32-s + ⋯

Functional equation

Λ(s)=(97344s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(97344s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 9734497344    =    26⋅32⋅1322^{6} \cdot 3^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 6.206736.20673
Root analytic conductor: 1.578391.57839
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 97344, ( :1/2,1/2), 1)(4,\ 97344,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.8495511601.849551160
L(12)L(\frac12) ≈\approx 1.8495511601.849551160
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}
3C1C_1 (1−T)2 ( 1 - T )^{2}
13C2C_2 1−4T+pT2 1 - 4 T + p T^{2}
good5C22C_2^2 1−4T+8T2−4pT3+p2T4 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4}
7C22C_2^2 1−6T+18T2−6pT3+p2T4 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4}
11C22C_2^2 1+8T+32T2+8pT3+p2T4 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4}
17C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
19C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4}
23C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
29C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
31C22C_2^2 1−2T+2T2−2pT3+p2T4 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4}
37C2C_2 (1+2T+pT2)(1+12T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} )
41C22C_2^2 1+8T+32T2+8pT3+p2T4 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4}
43C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
47C22C_2^2 1+4T+8T2+4pT3+p2T4 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4}
53C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
59C22C_2^2 1+4T+8T2+4pT3+p2T4 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4}
61C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
67C22C_2^2 1−6T+18T2−6pT3+p2T4 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4}
71C22C_2^2 1+8T+32T2+8pT3+p2T4 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4}
73C22C_2^2 1+2T+2T2+2pT3+p2T4 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
79C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
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+4T+8T2+4pT3+p2T4 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4}
97C22C_2^2 1−6T+18T2−6pT3+p2T4 1 - 6 T + 18 T^{2} - 6 p T^{3} + 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

−11.59140910960777246297133487129, −11.03843255949456336318029551461, −10.58753583476993340298298772515, −10.55842106458327365195658126946, −10.11946079920542564162983275817, −9.484086352400828709565420899307, −8.913492157933406423399219753868, −8.507378753893691547015285373015, −8.250438907965759386514389555351, −8.121972295425459131221965947213, −7.12150507910982425926081735349, −7.07977206422364817342323710938, −6.04111746428192283709525017706, −5.31889933963681531072655976191, −4.97239467948513122547504169858, −4.48217423587517232634617200200, −3.14297076464566174802901790178, −2.40687491424041976776897269724, −1.82542213834283294130118914955, −1.49509579743978856137702165840, 1.49509579743978856137702165840, 1.82542213834283294130118914955, 2.40687491424041976776897269724, 3.14297076464566174802901790178, 4.48217423587517232634617200200, 4.97239467948513122547504169858, 5.31889933963681531072655976191, 6.04111746428192283709525017706, 7.07977206422364817342323710938, 7.12150507910982425926081735349, 8.121972295425459131221965947213, 8.250438907965759386514389555351, 8.507378753893691547015285373015, 8.913492157933406423399219753868, 9.484086352400828709565420899307, 10.11946079920542564162983275817, 10.55842106458327365195658126946, 10.58753583476993340298298772515, 11.03843255949456336318029551461, 11.59140910960777246297133487129

Graph of the ZZ-function along the critical line