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Properties

Label 8-1152e4-1.1-c2e4-0-8
Degree 88
Conductor 1.761×10121.761\times 10^{12}
Sign 11
Analytic cond. 970845.970845.
Root an. cond. 5.602655.60265
Motivic weight 22
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  + 8·5-s + 24·13-s + 8·17-s + 4·25-s + 136·29-s − 40·37-s − 8·41-s + 100·49-s − 88·53-s − 296·61-s + 192·65-s + 88·73-s + 64·85-s − 216·89-s − 328·97-s − 24·101-s + 440·109-s + 312·113-s + 140·121-s − 72·125-s + 127-s + 131-s + 137-s + 139-s + 1.08e3·145-s + 149-s + 151-s + ⋯
L(s)  = 1  + 8/5·5-s + 1.84·13-s + 8/17·17-s + 4/25·25-s + 4.68·29-s − 1.08·37-s − 0.195·41-s + 2.04·49-s − 1.66·53-s − 4.85·61-s + 2.95·65-s + 1.20·73-s + 0.752·85-s − 2.42·89-s − 3.38·97-s − 0.237·101-s + 4.03·109-s + 2.76·113-s + 1.15·121-s − 0.575·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 7.50·145-s + 0.00671·149-s + 0.00662·151-s + ⋯

Functional equation

Λ(s)=((228⋅38)s/2 ΓC(s)4 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
Λ(s)=((228⋅38)s/2 ΓC(s+1)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 228⋅382^{28} \cdot 3^{8}
Sign: 11
Analytic conductor: 970845.970845.
Root analytic conductor: 5.602655.60265
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 228⋅38, ( :1,1,1,1), 1)(8,\ 2^{28} \cdot 3^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 4.2439856534.243985653
L(12)L(\frac12) ≈\approx 4.2439856534.243985653
L(2)L(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
3 1 1
good5D4D_{4} (1−4T+22T2−4p2T3+p4T4)2 ( 1 - 4 T + 22 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2}
7D4×C2D_4\times C_2 1−100T2+5254T4−100p4T6+p8T8 1 - 100 T^{2} + 5254 T^{4} - 100 p^{4} T^{6} + p^{8} T^{8}
11D4×C2D_4\times C_2 1−140T2+5382T4−140p4T6+p8T8 1 - 140 T^{2} + 5382 T^{4} - 140 p^{4} T^{6} + p^{8} T^{8}
13D4D_{4} (1−12T+342T2−12p2T3+p4T4)2 ( 1 - 12 T + 342 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2}
17D4D_{4} (1−4T+454T2−4p2T3+p4T4)2 ( 1 - 4 T + 454 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2}
19D4×C2D_4\times C_2 1−780T2+402374T4−780p4T6+p8T8 1 - 780 T^{2} + 402374 T^{4} - 780 p^{4} T^{6} + p^{8} T^{8}
23D4×C2D_4\times C_2 1−484T2+517894T4−484p4T6+p8T8 1 - 484 T^{2} + 517894 T^{4} - 484 p^{4} T^{6} + p^{8} T^{8}
29D4D_{4} (1−68T+2806T2−68p2T3+p4T4)2 ( 1 - 68 T + 2806 T^{2} - 68 p^{2} T^{3} + p^{4} T^{4} )^{2}
31C22C_2^2 (1+126T2+p4T4)2 ( 1 + 126 T^{2} + p^{4} T^{4} )^{2}
37D4D_{4} (1+20T+1270T2+20p2T3+p4T4)2 ( 1 + 20 T + 1270 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2}
41D4D_{4} (1+4T+2854T2+4p2T3+p4T4)2 ( 1 + 4 T + 2854 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2}
43D4×C2D_4\times C_2 1−2764T2+8710534T4−2764p4T6+p8T8 1 - 2764 T^{2} + 8710534 T^{4} - 2764 p^{4} T^{6} + p^{8} T^{8}
47D4×C2D_4\times C_2 1−7428T2+23258246T4−7428p4T6+p8T8 1 - 7428 T^{2} + 23258246 T^{4} - 7428 p^{4} T^{6} + p^{8} T^{8}
53D4D_{4} (1+44T+6070T2+44p2T3+p4T4)2 ( 1 + 44 T + 6070 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2}
59D4×C2D_4\times C_2 1−11020T2+52720774T4−11020p4T6+p8T8 1 - 11020 T^{2} + 52720774 T^{4} - 11020 p^{4} T^{6} + p^{8} T^{8}
61D4D_{4} (1+148T+11350T2+148p2T3+p4T4)2 ( 1 + 148 T + 11350 T^{2} + 148 p^{2} T^{3} + p^{4} T^{4} )^{2}
67D4×C2D_4\times C_2 1−11980T2+75342534T4−11980p4T6+p8T8 1 - 11980 T^{2} + 75342534 T^{4} - 11980 p^{4} T^{6} + p^{8} T^{8}
71D4×C2D_4\times C_2 1−18276T2+133865606T4−18276p4T6+p8T8 1 - 18276 T^{2} + 133865606 T^{4} - 18276 p^{4} T^{6} + p^{8} T^{8}
73D4D_{4} (1−44T+9990T2−44p2T3+p4T4)2 ( 1 - 44 T + 9990 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{2}
79D4×C2D_4\times C_2 1−1028T2+28324230T4−1028p4T6+p8T8 1 - 1028 T^{2} + 28324230 T^{4} - 1028 p^{4} T^{6} + p^{8} T^{8}
83D4×C2D_4\times C_2 1−5452T2+97966918T4−5452p4T6+p8T8 1 - 5452 T^{2} + 97966918 T^{4} - 5452 p^{4} T^{6} + p^{8} T^{8}
89D4D_{4} (1+108T+15558T2+108p2T3+p4T4)2 ( 1 + 108 T + 15558 T^{2} + 108 p^{2} T^{3} + p^{4} T^{4} )^{2}
97D4D_{4} (1+164T+22342T2+164p2T3+p4T4)2 ( 1 + 164 T + 22342 T^{2} + 164 p^{2} T^{3} + p^{4} 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

−6.81125283867734505342876355094, −6.61305073222305456755872921900, −6.22625227185466288804829429416, −6.05351852560833262930926439572, −5.88500301737042743491089433190, −5.88426142744681422413003726991, −5.55108766592641729578118744814, −5.44844050455904940744911690523, −4.74558532804764279152745288850, −4.71867040930663527999360604692, −4.57283466723973742930082598553, −4.54747047727654709425165837113, −4.02866305823115260208026311973, −3.72953669468041300617525627341, −3.21816053225964689935714969684, −3.20836054977621608618266975002, −3.12690653441728753669752836101, −2.71064459419176726937569400257, −2.22552855246060675926367930578, −2.14941232133283821755671965146, −1.74243779977826900137093323558, −1.26569773991839882038636133384, −1.08885056622549555231427795650, −1.06390857272983656068718462499, −0.23695901959675035100910703579, 0.23695901959675035100910703579, 1.06390857272983656068718462499, 1.08885056622549555231427795650, 1.26569773991839882038636133384, 1.74243779977826900137093323558, 2.14941232133283821755671965146, 2.22552855246060675926367930578, 2.71064459419176726937569400257, 3.12690653441728753669752836101, 3.20836054977621608618266975002, 3.21816053225964689935714969684, 3.72953669468041300617525627341, 4.02866305823115260208026311973, 4.54747047727654709425165837113, 4.57283466723973742930082598553, 4.71867040930663527999360604692, 4.74558532804764279152745288850, 5.44844050455904940744911690523, 5.55108766592641729578118744814, 5.88426142744681422413003726991, 5.88500301737042743491089433190, 6.05351852560833262930926439572, 6.22625227185466288804829429416, 6.61305073222305456755872921900, 6.81125283867734505342876355094

Graph of the ZZ-function along the critical line