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Properties

Label 8-416e4-1.1-c1e4-0-1
Degree 88
Conductor 2994837913629948379136
Sign 11
Analytic cond. 121.753121.753
Root an. cond. 1.822571.82257
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  + 4·3-s − 2·9-s − 4·11-s − 8·19-s − 40·27-s − 16·33-s + 24·41-s − 32·57-s − 32·59-s − 12·67-s − 24·73-s − 55·81-s − 20·83-s − 8·89-s − 28·97-s + 8·99-s + 8·107-s + 16·113-s + 8·121-s + 96·123-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  + 2.30·3-s − 2/3·9-s − 1.20·11-s − 1.83·19-s − 7.69·27-s − 2.78·33-s + 3.74·41-s − 4.23·57-s − 4.16·59-s − 1.46·67-s − 2.80·73-s − 6.11·81-s − 2.19·83-s − 0.847·89-s − 2.84·97-s + 0.804·99-s + 0.773·107-s + 1.50·113-s + 8/11·121-s + 8.65·123-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

Λ(s)=((220⋅134)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((220⋅134)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 220⋅1342^{20} \cdot 13^{4}
Sign: 11
Analytic conductor: 121.753121.753
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 220⋅134, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{20} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.50389977570.5038997757
L(12)L(\frac12) ≈\approx 0.50389977570.5038997757
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)
bad2 1 1
13C22C_2^2 1+p2T4 1 + p^{2} T^{4}
good3C2C_2 (1−T+pT2)4 ( 1 - T + p T^{2} )^{4}
5C23C_2^3 1−41T4+p4T8 1 - 41 T^{4} + p^{4} T^{8}
7C23C_2^3 1−97T4+p4T8 1 - 97 T^{4} + p^{4} T^{8}
11C22C_2^2 (1+2T+2T2+2pT3+p2T4)2 ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
17C22C_2^2 (1−25T2+p2T4)2 ( 1 - 25 T^{2} + p^{2} T^{4} )^{2}
19C22C_2^2 (1+4T+8T2+4pT3+p2T4)2 ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
23C22C_2^2 (1+20T2+p2T4)2 ( 1 + 20 T^{2} + p^{2} T^{4} )^{2}
29C22C_2^2 (1−32T2+p2T4)2 ( 1 - 32 T^{2} + p^{2} T^{4} )^{2}
31C22C_2^2×\timesC22C_2^2 (1−12T+72T2−12pT3+p2T4)(1+12T+72T2+12pT3+p2T4) ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )
37C23C_2^3 1+983T4+p4T8 1 + 983 T^{4} + p^{4} T^{8}
41C22C_2^2 (1−12T+72T2−12pT3+p2T4)2 ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}
43C22C_2^2 (1−85T2+p2T4)2 ( 1 - 85 T^{2} + p^{2} T^{4} )^{2}
47C23C_2^3 1+2143T4+p4T8 1 + 2143 T^{4} + p^{4} T^{8}
53C22C_2^2 (1−80T2+p2T4)2 ( 1 - 80 T^{2} + p^{2} T^{4} )^{2}
59C22C_2^2 (1+16T+128T2+16pT3+p2T4)2 ( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2}
61C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
67C22C_2^2 (1+6T+18T2+6pT3+p2T4)2 ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}
71C23C_2^3 1−9457T4+p4T8 1 - 9457 T^{4} + p^{4} T^{8}
73C22C_2^2 (1+12T+72T2+12pT3+p2T4)2 ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}
79C22C_2^2 (1−132T2+p2T4)2 ( 1 - 132 T^{2} + p^{2} T^{4} )^{2}
83C22C_2^2 (1+10T+50T2+10pT3+p2T4)2 ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2}
89C22C_2^2 (1+4T+8T2+4pT3+p2T4)2 ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
97C22C_2^2 (1+14T+98T2+14pT3+p2T4)2 ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2}
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   L(s)=∏p ∏j=18(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.183503863471162482817469421664, −7.949970144349250383813649320626, −7.71442207920456603963892291435, −7.60721889227404273118802203977, −7.37186392693435805117818798283, −7.14328859976291820371974682354, −6.53306096055565267878808075195, −6.29858617194998800922041737282, −5.98747868371912356386435455814, −5.81671496112109025206349593107, −5.68900522642042334314257742942, −5.56051991924676673029888604411, −4.99132920166081112136667617576, −4.44603574149262605827057381643, −4.44283662951956588683922956516, −4.21350629763050392852135877784, −3.57496313961042972605135272492, −3.48804977272946765309609836342, −2.93739711415506004510821247999, −2.67146770059946876187202292332, −2.65575089887556229422964292453, −2.63960168777552721901613107640, −1.83594166905495058642333256099, −1.64159201182266667141281234590, −0.19767752283746219961372471541, 0.19767752283746219961372471541, 1.64159201182266667141281234590, 1.83594166905495058642333256099, 2.63960168777552721901613107640, 2.65575089887556229422964292453, 2.67146770059946876187202292332, 2.93739711415506004510821247999, 3.48804977272946765309609836342, 3.57496313961042972605135272492, 4.21350629763050392852135877784, 4.44283662951956588683922956516, 4.44603574149262605827057381643, 4.99132920166081112136667617576, 5.56051991924676673029888604411, 5.68900522642042334314257742942, 5.81671496112109025206349593107, 5.98747868371912356386435455814, 6.29858617194998800922041737282, 6.53306096055565267878808075195, 7.14328859976291820371974682354, 7.37186392693435805117818798283, 7.60721889227404273118802203977, 7.71442207920456603963892291435, 7.949970144349250383813649320626, 8.183503863471162482817469421664

Graph of the ZZ-function along the critical line