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Properties

Label 8-270e4-1.1-c2e4-0-0
Degree 88
Conductor 53144100005314410000
Sign 11
Analytic cond. 2929.512929.51
Root an. cond. 2.712372.71237
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  + 4·4-s − 12·5-s + 12·16-s + 12·17-s − 12·19-s − 48·20-s − 60·23-s + 66·25-s − 68·31-s + 240·47-s + 8·49-s − 204·53-s − 196·61-s + 32·64-s + 48·68-s − 48·76-s + 180·79-s − 144·80-s − 108·83-s − 144·85-s − 240·92-s + 144·95-s + 264·100-s + 144·107-s − 76·109-s − 48·113-s + 720·115-s + ⋯
L(s)  = 1  + 4-s − 2.39·5-s + 3/4·16-s + 0.705·17-s − 0.631·19-s − 2.39·20-s − 2.60·23-s + 2.63·25-s − 2.19·31-s + 5.10·47-s + 8/49·49-s − 3.84·53-s − 3.21·61-s + 1/2·64-s + 0.705·68-s − 0.631·76-s + 2.27·79-s − 9/5·80-s − 1.30·83-s − 1.69·85-s − 2.60·92-s + 1.51·95-s + 2.63·100-s + 1.34·107-s − 0.697·109-s − 0.424·113-s + 6.26·115-s + ⋯

Functional equation

Λ(s)=((24⋅312⋅54)s/2 ΓC(s)4 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
Λ(s)=((24⋅312⋅54)s/2 ΓC(s+1)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 24⋅312⋅542^{4} \cdot 3^{12} \cdot 5^{4}
Sign: 11
Analytic conductor: 2929.512929.51
Root analytic conductor: 2.712372.71237
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 24⋅312⋅54, ( :1,1,1,1), 1)(8,\ 2^{4} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.0030931190920.003093119092
L(12)L(\frac12) ≈\approx 0.0030931190920.003093119092
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)
bad2C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
3 1 1
5D4D_{4} 1+12T+78T2+12p2T3+p4T4 1 + 12 T + 78 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4}
good7C22≀C2C_2^2 \wr C_2 1−8T2−3894T4−8p4T6+p8T8 1 - 8 T^{2} - 3894 T^{4} - 8 p^{4} T^{6} + p^{8} T^{8}
11C22≀C2C_2^2 \wr C_2 1−96T2+29786T4−96p4T6+p8T8 1 - 96 T^{2} + 29786 T^{4} - 96 p^{4} T^{6} + p^{8} T^{8}
13C22≀C2C_2^2 \wr C_2 1−352T2+71898T4−352p4T6+p8T8 1 - 352 T^{2} + 71898 T^{4} - 352 p^{4} T^{6} + p^{8} T^{8}
17D4D_{4} (1−6T+489T2−6p2T3+p4T4)2 ( 1 - 6 T + 489 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2}
19D4D_{4} (1+6T+713T2+6p2T3+p4T4)2 ( 1 + 6 T + 713 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{2}
23D4D_{4} (1+30T+891T2+30p2T3+p4T4)2 ( 1 + 30 T + 891 T^{2} + 30 p^{2} T^{3} + p^{4} T^{4} )^{2}
29C22≀C2C_2^2 \wr C_2 1−1096T2+1242474T4−1096p4T6+p8T8 1 - 1096 T^{2} + 1242474 T^{4} - 1096 p^{4} T^{6} + p^{8} T^{8}
31D4D_{4} (1+34T+1761T2+34p2T3+p4T4)2 ( 1 + 34 T + 1761 T^{2} + 34 p^{2} T^{3} + p^{4} T^{4} )^{2}
37C22≀C2C_2^2 \wr C_2 1−68T2−1268634T4−68p4T6+p8T8 1 - 68 T^{2} - 1268634 T^{4} - 68 p^{4} T^{6} + p^{8} T^{8}
41C22≀C2C_2^2 \wr C_2 1−5352T2+12407498T4−5352p4T6+p8T8 1 - 5352 T^{2} + 12407498 T^{4} - 5352 p^{4} T^{6} + p^{8} T^{8}
43C22≀C2C_2^2 \wr C_2 1−5672T2+14203050T4−5672p4T6+p8T8 1 - 5672 T^{2} + 14203050 T^{4} - 5672 p^{4} T^{6} + p^{8} T^{8}
47D4D_{4} (1−120T+7626T2−120p2T3+p4T4)2 ( 1 - 120 T + 7626 T^{2} - 120 p^{2} T^{3} + p^{4} T^{4} )^{2}
53D4D_{4} (1+102T+8057T2+102p2T3+p4T4)2 ( 1 + 102 T + 8057 T^{2} + 102 p^{2} T^{3} + p^{4} T^{4} )^{2}
59C22≀C2C_2^2 \wr C_2 1−10848T2+53616410T4−10848p4T6+p8T8 1 - 10848 T^{2} + 53616410 T^{4} - 10848 p^{4} T^{6} + p^{8} T^{8}
61D4D_{4} (1+98T+8691T2+98p2T3+p4T4)2 ( 1 + 98 T + 8691 T^{2} + 98 p^{2} T^{3} + p^{4} T^{4} )^{2}
67C22≀C2C_2^2 \wr C_2 1+6508T2+32310150T4+6508p4T6+p8T8 1 + 6508 T^{2} + 32310150 T^{4} + 6508 p^{4} T^{6} + p^{8} T^{8}
71C22≀C2C_2^2 \wr C_2 1−4288T2+45932730T4−4288p4T6+p8T8 1 - 4288 T^{2} + 45932730 T^{4} - 4288 p^{4} T^{6} + p^{8} T^{8}
73C22≀C2C_2^2 \wr C_2 1−13280T2+84777594T4−13280p4T6+p8T8 1 - 13280 T^{2} + 84777594 T^{4} - 13280 p^{4} T^{6} + p^{8} T^{8}
79D4D_{4} (1−90T+6569T2−90p2T3+p4T4)2 ( 1 - 90 T + 6569 T^{2} - 90 p^{2} T^{3} + p^{4} T^{4} )^{2}
83D4D_{4} (1+54T+7779T2+54p2T3+p4T4)2 ( 1 + 54 T + 7779 T^{2} + 54 p^{2} T^{3} + p^{4} T^{4} )^{2}
89C22≀C2C_2^2 \wr C_2 1−8136T2+91108874T4−8136p4T6+p8T8 1 - 8136 T^{2} + 91108874 T^{4} - 8136 p^{4} T^{6} + p^{8} T^{8}
97C22≀C2C_2^2 \wr C_2 1−19172T2+267913158T4−19172p4T6+p8T8 1 - 19172 T^{2} + 267913158 T^{4} - 19172 p^{4} T^{6} + p^{8} T^{8}
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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

−8.417304005703146287284569980597, −8.062347681090796004110683855964, −7.72587265447779194131888066790, −7.70160775134969670698516823934, −7.51181083065864069572447934547, −7.35781161800555583840081507828, −7.05306337822045578066539765849, −6.61995464469389639668984929066, −6.23497802634086679275257256508, −6.18683368950351144786000148334, −5.81820864562338701314150277122, −5.47586346339973710203501027583, −5.35918875587960436062511752831, −4.55627242451186463850779747897, −4.49266772781826075664787255832, −4.19778619062035900273620195074, −3.94615722225874517332502935570, −3.46828571321495591781688823344, −3.42304134135874615096236568517, −3.03322616900163626894039551733, −2.39408445534723344321305199832, −2.09405665828087729871698837057, −1.65220395114889609306075717394, −0.963257331480092248079427387955, −0.01546708143175349743667653317, 0.01546708143175349743667653317, 0.963257331480092248079427387955, 1.65220395114889609306075717394, 2.09405665828087729871698837057, 2.39408445534723344321305199832, 3.03322616900163626894039551733, 3.42304134135874615096236568517, 3.46828571321495591781688823344, 3.94615722225874517332502935570, 4.19778619062035900273620195074, 4.49266772781826075664787255832, 4.55627242451186463850779747897, 5.35918875587960436062511752831, 5.47586346339973710203501027583, 5.81820864562338701314150277122, 6.18683368950351144786000148334, 6.23497802634086679275257256508, 6.61995464469389639668984929066, 7.05306337822045578066539765849, 7.35781161800555583840081507828, 7.51181083065864069572447934547, 7.70160775134969670698516823934, 7.72587265447779194131888066790, 8.062347681090796004110683855964, 8.417304005703146287284569980597

Graph of the ZZ-function along the critical line