Properties

Label 8-3332e4-1.1-c0e4-0-6
Degree 88
Conductor 1.233×10141.233\times 10^{14}
Sign 11
Analytic cond. 7.646247.64624
Root an. cond. 1.289521.28952
Motivic weight 00
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  − 16-s − 2·25-s + 4·37-s + 4·41-s − 4·53-s + 4·61-s + 8·101-s − 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  − 16-s − 2·25-s + 4·37-s + 4·41-s − 4·53-s + 4·61-s + 8·101-s − 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯

Functional equation

Λ(s)=((2878174)s/2ΓC(s)4L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((2878174)s/2ΓC(s)4L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 28781742^{8} \cdot 7^{8} \cdot 17^{4}
Sign: 11
Analytic conductor: 7.646247.64624
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 2878174, ( :0,0,0,0), 1)(8,\ 2^{8} \cdot 7^{8} \cdot 17^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) \approx 1.7328427171.732842717
L(12)L(\frac12) \approx 1.7328427171.732842717
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C22C_2^2 1+T4 1 + T^{4}
7 1 1
17C22C_2^2 1+T4 1 + T^{4}
good3C4×C2C_4\times C_2 1+T8 1 + T^{8}
5C2C_2×\timesC22C_2^2 (1+T2)2(1+T4) ( 1 + T^{2} )^{2}( 1 + T^{4} )
11C4×C2C_4\times C_2 1+T8 1 + T^{8}
13C1C_1×\timesC1C_1 (1T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
19C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
23C4×C2C_4\times C_2 1+T8 1 + T^{8}
29C2C_2×\timesC22C_2^2 (1+T2)2(1+T4) ( 1 + T^{2} )^{2}( 1 + T^{4} )
31C4×C2C_4\times C_2 1+T8 1 + T^{8}
37C1C_1×\timesC22C_2^2 (1T)4(1+T4) ( 1 - T )^{4}( 1 + T^{4} )
41C1C_1×\timesC22C_2^2 (1T)4(1+T4) ( 1 - T )^{4}( 1 + T^{4} )
43C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
47C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
53C1C_1×\timesC2C_2 (1+T)4(1+T2)2 ( 1 + T )^{4}( 1 + T^{2} )^{2}
59C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
61C1C_1×\timesC22C_2^2 (1T)4(1+T4) ( 1 - T )^{4}( 1 + T^{4} )
67C1C_1×\timesC1C_1 (1T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
71C4×C2C_4\times C_2 1+T8 1 + T^{8}
73C2C_2×\timesC22C_2^2 (1+T2)2(1+T4) ( 1 + T^{2} )^{2}( 1 + T^{4} )
79C4×C2C_4\times C_2 1+T8 1 + T^{8}
83C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
89C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
97C2C_2×\timesC22C_2^2 (1+T2)2(1+T4) ( 1 + T^{2} )^{2}( 1 + T^{4} )
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   L(s)=p j=18(1αj,pps)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

−6.38074750558724713506387760046, −5.99954975419966063869477084708, −5.92382771393812870142007529452, −5.77978260182117092063982908069, −5.58335395566535234691631687726, −5.40727290182896558775355125325, −4.95418161849566728379002367075, −4.79402301021264576183173810456, −4.76753043535797753058362060340, −4.34711649536263176711536211360, −4.21932873224493985530803065026, −4.14120816936939216910316474815, −3.97289982351400572450958247198, −3.66746285278900886454033567017, −3.32725907116285632353854040213, −3.07206091283863621809561178184, −2.95020549410106364208670631832, −2.62417097887203082597183717696, −2.21442304092380673199566575360, −2.18186718006145966775075437264, −2.16111172548121158707996751811, −1.66724438166004028089889012926, −1.19184740120769111915557366517, −0.821717434010725781486152411641, −0.65721834447309321850082619888, 0.65721834447309321850082619888, 0.821717434010725781486152411641, 1.19184740120769111915557366517, 1.66724438166004028089889012926, 2.16111172548121158707996751811, 2.18186718006145966775075437264, 2.21442304092380673199566575360, 2.62417097887203082597183717696, 2.95020549410106364208670631832, 3.07206091283863621809561178184, 3.32725907116285632353854040213, 3.66746285278900886454033567017, 3.97289982351400572450958247198, 4.14120816936939216910316474815, 4.21932873224493985530803065026, 4.34711649536263176711536211360, 4.76753043535797753058362060340, 4.79402301021264576183173810456, 4.95418161849566728379002367075, 5.40727290182896558775355125325, 5.58335395566535234691631687726, 5.77978260182117092063982908069, 5.92382771393812870142007529452, 5.99954975419966063869477084708, 6.38074750558724713506387760046

Graph of the ZZ-function along the critical line