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Properties

Label 8-1920e4-1.1-c0e4-0-2
Degree 88
Conductor 135895.450×108135895.450\times 10^{8}
Sign 11
Analytic cond. 0.8430110.843011
Root an. cond. 0.9788790.978879
Motivic weight 00
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  + 4·19-s − 4·49-s − 4·61-s + 8·79-s − 81-s + 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  + 4·19-s − 4·49-s − 4·61-s + 8·79-s − 81-s + 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯

Functional equation

Λ(s)=((228⋅34⋅54)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((228⋅34⋅54)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 228⋅34⋅542^{28} \cdot 3^{4} \cdot 5^{4}
Sign: 11
Analytic conductor: 0.8430110.843011
Root analytic conductor: 0.9788790.978879
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 228⋅34⋅54, ( :0,0,0,0), 1)(8,\ 2^{28} \cdot 3^{4} \cdot 5^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.5404523461.540452346
L(12)L(\frac12) ≈\approx 1.5404523461.540452346
L(1)L(1) 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
3C22C_2^2 1+T4 1 + T^{4}
5C22C_2^2 1+T4 1 + T^{4}
good7C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
11C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
13C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
17C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
19C1C_1×\timesC2C_2 (1−T)4(1+T2)2 ( 1 - T )^{4}( 1 + T^{2} )^{2}
23C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
29C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
31C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
37C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
41C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
43C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
47C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
53C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
59C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
61C1C_1×\timesC2C_2 (1+T)4(1+T2)2 ( 1 + T )^{4}( 1 + T^{2} )^{2}
67C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
71C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
73C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
79C1C_1 (1−T)8 ( 1 - T )^{8}
83C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
89C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
97C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
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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

−6.70403297876323933341092950439, −6.60406978827796684763351198882, −6.22870055801824354069459311506, −6.15482831036092311680169594509, −6.02832272998309290544903564838, −5.53044713783399615094760808943, −5.48446583179660042208604885098, −5.32846993092474744399879322591, −4.93037100735730426237105588860, −4.89964597872888503742949498674, −4.66616075109678715808851924896, −4.54147855639560133824732523319, −4.21035481134347670454651788897, −3.72322816787976880019680435729, −3.57781372830606672855267773889, −3.25647727923477409085927961514, −3.13451694512454482195420610889, −3.08555575098576558348423473765, −2.98967766695846537264953587272, −2.17186569299943386037840278972, −1.95913707563857676612491659535, −1.95778654842782115418464481243, −1.38653300217874830927539256674, −1.02629549577599849375829471900, −0.77884833184188450067356919233, 0.77884833184188450067356919233, 1.02629549577599849375829471900, 1.38653300217874830927539256674, 1.95778654842782115418464481243, 1.95913707563857676612491659535, 2.17186569299943386037840278972, 2.98967766695846537264953587272, 3.08555575098576558348423473765, 3.13451694512454482195420610889, 3.25647727923477409085927961514, 3.57781372830606672855267773889, 3.72322816787976880019680435729, 4.21035481134347670454651788897, 4.54147855639560133824732523319, 4.66616075109678715808851924896, 4.89964597872888503742949498674, 4.93037100735730426237105588860, 5.32846993092474744399879322591, 5.48446583179660042208604885098, 5.53044713783399615094760808943, 6.02832272998309290544903564838, 6.15482831036092311680169594509, 6.22870055801824354069459311506, 6.60406978827796684763351198882, 6.70403297876323933341092950439

Graph of the ZZ-function along the critical line