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Properties

Label 8-384e4-1.1-c4e4-0-2
Degree 88
Conductor 2174327193621743271936
Sign 11
Analytic cond. 2.48257×1062.48257\times 10^{6}
Root an. cond. 6.300326.30032
Motivic weight 44
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  + 54·9-s − 1.35e3·17-s + 676·25-s + 2.31e3·41-s + 4.13e3·49-s + 3.49e4·73-s + 2.18e3·81-s + 3.64e3·89-s + 2.16e4·97-s − 1.26e4·113-s − 4.23e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 7.30e4·153-s + 157-s + 163-s + 167-s + 1.06e5·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 2/3·9-s − 4.67·17-s + 1.08·25-s + 1.37·41-s + 1.72·49-s + 6.55·73-s + 1/3·81-s + 0.459·89-s + 2.30·97-s − 0.991·113-s − 2.89·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s − 3.11·153-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 3.74·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + ⋯

Functional equation

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

Invariants

Degree: 88
Conductor: 228⋅342^{28} \cdot 3^{4}
Sign: 11
Analytic conductor: 2.48257×1062.48257\times 10^{6}
Root analytic conductor: 6.300326.30032
Motivic weight: 44
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 228⋅34, ( :2,2,2,2), 1)(8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 2, 2, 2, 2 ),\ 1 )

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 3.2881697783.288169778
L(12)L(\frac12) ≈\approx 3.2881697783.288169778
L(3)L(3) 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
3C2C_2 (1−p3T2)2 ( 1 - p^{3} T^{2} )^{2}
good5C22C_2^2 (1−338T2+p8T4)2 ( 1 - 338 T^{2} + p^{8} T^{4} )^{2}
7C22C_2^2 (1−2066T2+p8T4)2 ( 1 - 2066 T^{2} + p^{8} T^{4} )^{2}
11C22C_2^2 (1+21170T2+p8T4)2 ( 1 + 21170 T^{2} + p^{8} T^{4} )^{2}
13C22C_2^2 (1−53474T2+p8T4)2 ( 1 - 53474 T^{2} + p^{8} T^{4} )^{2}
17C2C_2 (1+338T+p4T2)4 ( 1 + 338 T + p^{4} T^{2} )^{4}
19C22C_2^2 (1+260594T2+p8T4)2 ( 1 + 260594 T^{2} + p^{8} T^{4} )^{2}
23C22C_2^2 (1−23426T2+p8T4)2 ( 1 - 23426 T^{2} + p^{8} T^{4} )^{2}
29C22C_2^2 (1+271726T2+p8T4)2 ( 1 + 271726 T^{2} + p^{8} T^{4} )^{2}
31C22C_2^2 (1−137042T2+p8T4)2 ( 1 - 137042 T^{2} + p^{8} T^{4} )^{2}
37C22C_2^2 (1−3689954T2+p8T4)2 ( 1 - 3689954 T^{2} + p^{8} T^{4} )^{2}
41C2C_2 (1−578T+p4T2)4 ( 1 - 578 T + p^{4} T^{2} )^{4}
43C22C_2^2 (1+2716850T2+p8T4)2 ( 1 + 2716850 T^{2} + p^{8} T^{4} )^{2}
47C22C_2^2 (1−4933058T2+p8T4)2 ( 1 - 4933058 T^{2} + p^{8} T^{4} )^{2}
53C22C_2^2 (1−9797330T2+p8T4)2 ( 1 - 9797330 T^{2} + p^{8} T^{4} )^{2}
59C22C_2^2 (1+22798130T2+p8T4)2 ( 1 + 22798130 T^{2} + p^{8} T^{4} )^{2}
61C2C_2 (1−3794T+p4T2)2(1+3794T+p4T2)2 ( 1 - 3794 T + p^{4} T^{2} )^{2}( 1 + 3794 T + p^{4} T^{2} )^{2}
67C22C_2^2 (1−28013710T2+p8T4)2 ( 1 - 28013710 T^{2} + p^{8} T^{4} )^{2}
71C22C_2^2 (1−32426498T2+p8T4)2 ( 1 - 32426498 T^{2} + p^{8} T^{4} )^{2}
73C2C_2 (1−8734T+p4T2)4 ( 1 - 8734 T + p^{4} T^{2} )^{4}
79C22C_2^2 (1+48571438T2+p8T4)2 ( 1 + 48571438 T^{2} + p^{8} T^{4} )^{2}
83C22C_2^2 (1−79276558T2+p8T4)2 ( 1 - 79276558 T^{2} + p^{8} T^{4} )^{2}
89C2C_2 (1−910T+p4T2)4 ( 1 - 910 T + p^{4} T^{2} )^{4}
97C2C_2 (1−5422T+p4T2)4 ( 1 - 5422 T + p^{4} 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

−7.58949365398757466259232689807, −7.46166024684715774792032001736, −6.77871611233654619397732711077, −6.67771946087191592682582015915, −6.58416583268060051005450190116, −6.49968589366631493045522382732, −6.37140804535537813900463707570, −5.67879865361347589743628666262, −5.49907790169617691126044761718, −5.04439148759725351036286504265, −5.00862572319347721066005396858, −4.63183664291523399896086026176, −4.40882115644136230983390299240, −4.03293527824268505358525342081, −3.86828312726056042266193906857, −3.79538142655755248563776694299, −3.04755833984427561508606737221, −2.76306106360925704755386886234, −2.34428443615831942889709409196, −2.13691012841414052899109094456, −2.09126214270688890135136915585, −1.51484997005490701423547336672, −0.817161908041554561167793361454, −0.69410615792173018095000668515, −0.29137577582836413102053743548, 0.29137577582836413102053743548, 0.69410615792173018095000668515, 0.817161908041554561167793361454, 1.51484997005490701423547336672, 2.09126214270688890135136915585, 2.13691012841414052899109094456, 2.34428443615831942889709409196, 2.76306106360925704755386886234, 3.04755833984427561508606737221, 3.79538142655755248563776694299, 3.86828312726056042266193906857, 4.03293527824268505358525342081, 4.40882115644136230983390299240, 4.63183664291523399896086026176, 5.00862572319347721066005396858, 5.04439148759725351036286504265, 5.49907790169617691126044761718, 5.67879865361347589743628666262, 6.37140804535537813900463707570, 6.49968589366631493045522382732, 6.58416583268060051005450190116, 6.67771946087191592682582015915, 6.77871611233654619397732711077, 7.46166024684715774792032001736, 7.58949365398757466259232689807

Graph of the ZZ-function along the critical line