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Properties

Label 8-2400e4-1.1-c1e4-0-20
Degree 88
Conductor 3.318×10133.318\times 10^{13}
Sign 11
Analytic cond. 134881.134881.
Root an. cond. 4.377684.37768
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  + 2·3-s + 3·9-s + 4·19-s + 10·27-s − 40·43-s + 28·49-s + 8·57-s − 28·67-s + 4·73-s + 20·81-s + 40·97-s − 14·121-s + 127-s − 80·129-s + 131-s + 137-s + 139-s + 56·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 12·171-s + 173-s + 179-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s + 0.917·19-s + 1.92·27-s − 6.09·43-s + 4·49-s + 1.05·57-s − 3.42·67-s + 0.468·73-s + 20/9·81-s + 4.06·97-s − 1.27·121-s + 0.0887·127-s − 7.04·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 4.61·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.917·171-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

Λ(s)=((220⋅34⋅58)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((220⋅34⋅58)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 220⋅34⋅582^{20} \cdot 3^{4} \cdot 5^{8}
Sign: 11
Analytic conductor: 134881.134881.
Root analytic conductor: 4.377684.37768
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 220⋅34⋅58, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{20} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 6.7098236836.709823683
L(12)L(\frac12) ≈\approx 6.7098236836.709823683
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
3C22C_2^2 1−2T+T2−2pT3+p2T4 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4}
5 1 1
good7C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
11C22C_2^2×\timesC22C_2^2 (1−6T+25T2−6pT3+p2T4)(1+6T+25T2+6pT3+p2T4) ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )
13C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
17C22C_2^2×\timesC22C_2^2 (1−6T+19T2−6pT3+p2T4)(1+6T+19T2+6pT3+p2T4) ( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} )
19C22C_2^2 (1−2T−15T2−2pT3+p2T4)2 ( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}
23C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
29C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
31C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
37C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
41C22C_2^2×\timesC22C_2^2 (1−6T−5T2−6pT3+p2T4)(1+6T−5T2+6pT3+p2T4) ( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} )
43C2C_2 (1+10T+pT2)4 ( 1 + 10 T + p T^{2} )^{4}
47C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
53C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
59C2C_2 (1−6T+pT2)2(1+6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2}
61C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
67C22C_2^2 (1+14T+129T2+14pT3+p2T4)2 ( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2}
71C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
73C22C_2^2 (1−2T−69T2−2pT3+p2T4)2 ( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}
79C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
83C22C_2^2×\timesC22C_2^2 (1−18T+241T2−18pT3+p2T4)(1+18T+241T2+18pT3+p2T4) ( 1 - 18 T + 241 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 241 T^{2} + 18 p T^{3} + p^{2} T^{4} )
89C22C_2^2×\timesC22C_2^2 (1−18T+235T2−18pT3+p2T4)(1+18T+235T2+18pT3+p2T4) ( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} )
97C2C_2 (1−10T+pT2)4 ( 1 - 10 T + p T^{2} )^{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

−6.29810652928244661522904869651, −6.23583169142696017185866501008, −6.04309951234332684605495159415, −5.64228958804215514425690321768, −5.61648469931404268091350290471, −5.21243116302503239715992755814, −4.95728637145093159668528647275, −4.94986362810558856804720390121, −4.67956448623710692734432629639, −4.58871857086127085476298681174, −4.20759989122704855256178912501, −3.78296992102574489846456675371, −3.76051372594321453718201247905, −3.51420331642559724177430194377, −3.43319629024432813609016860739, −2.92631142086972117996289086165, −2.84611926641799227065955828054, −2.75574903041701673804387748428, −2.26856364027439428353085544349, −2.11014474745977619101197590496, −1.63666091268367869758796457189, −1.42563044213681355331281555974, −1.40171065968265319313764330264, −0.64392262530148553165785526353, −0.44265595158499176382231464661, 0.44265595158499176382231464661, 0.64392262530148553165785526353, 1.40171065968265319313764330264, 1.42563044213681355331281555974, 1.63666091268367869758796457189, 2.11014474745977619101197590496, 2.26856364027439428353085544349, 2.75574903041701673804387748428, 2.84611926641799227065955828054, 2.92631142086972117996289086165, 3.43319629024432813609016860739, 3.51420331642559724177430194377, 3.76051372594321453718201247905, 3.78296992102574489846456675371, 4.20759989122704855256178912501, 4.58871857086127085476298681174, 4.67956448623710692734432629639, 4.94986362810558856804720390121, 4.95728637145093159668528647275, 5.21243116302503239715992755814, 5.61648469931404268091350290471, 5.64228958804215514425690321768, 6.04309951234332684605495159415, 6.23583169142696017185866501008, 6.29810652928244661522904869651

Graph of the ZZ-function along the critical line