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Properties

Label 8-24e8-1.1-c2e4-0-3
Degree 88
Conductor 110075314176110075314176
Sign 11
Analytic cond. 60677.860677.8
Root an. cond. 3.961673.96167
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  − 120·17-s + 76·25-s + 24·41-s − 4·49-s − 344·73-s − 312·89-s + 248·97-s + 24·113-s − 100·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 580·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 7.05·17-s + 3.03·25-s + 0.585·41-s − 0.0816·49-s − 4.71·73-s − 3.50·89-s + 2.55·97-s + 0.212·113-s − 0.826·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.43·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯

Functional equation

Λ(s)=((224⋅38)s/2 ΓC(s)4 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
Λ(s)=((224⋅38)s/2 ΓC(s+1)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 224⋅382^{24} \cdot 3^{8}
Sign: 11
Analytic conductor: 60677.860677.8
Root analytic conductor: 3.961673.96167
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 224⋅38, ( :1,1,1,1), 1)(8,\ 2^{24} \cdot 3^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.29352832450.2935283245
L(12)L(\frac12) ≈\approx 0.29352832450.2935283245
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
good5C22C_2^2 (1−38T2+p4T4)2 ( 1 - 38 T^{2} + p^{4} T^{4} )^{2}
7C22C_2^2 (1+2T2+p4T4)2 ( 1 + 2 T^{2} + p^{4} T^{4} )^{2}
11C22C_2^2 (1+50T2+p4T4)2 ( 1 + 50 T^{2} + p^{4} T^{4} )^{2}
13C22C_2^2 (1−290T2+p4T4)2 ( 1 - 290 T^{2} + p^{4} T^{4} )^{2}
17C2C_2 (1+30T+p2T2)4 ( 1 + 30 T + p^{2} T^{2} )^{4}
19C22C_2^2 (1+674T2+p4T4)2 ( 1 + 674 T^{2} + p^{4} T^{4} )^{2}
23C22C_2^2 (1−914T2+p4T4)2 ( 1 - 914 T^{2} + p^{4} T^{4} )^{2}
29C22C_2^2 (1+1018T2+p4T4)2 ( 1 + 1018 T^{2} + p^{4} T^{4} )^{2}
31C22C_2^2 (1−1726T2+p4T4)2 ( 1 - 1726 T^{2} + p^{4} T^{4} )^{2}
37C22C_2^2 (1+334T2+p4T4)2 ( 1 + 334 T^{2} + p^{4} T^{4} )^{2}
41C2C_2 (1−6T+p2T2)4 ( 1 - 6 T + p^{2} T^{2} )^{4}
43C22C_2^2 (1−190T2+p4T4)2 ( 1 - 190 T^{2} + p^{4} T^{4} )^{2}
47C22C_2^2 (1+2638T2+p4T4)2 ( 1 + 2638 T^{2} + p^{4} T^{4} )^{2}
53C22C_2^2 (1−5318T2+p4T4)2 ( 1 - 5318 T^{2} + p^{4} T^{4} )^{2}
59C22C_2^2 (1+3074T2+p4T4)2 ( 1 + 3074 T^{2} + p^{4} T^{4} )^{2}
61C2C_2 (1−74T+p2T2)2(1+74T+p2T2)2 ( 1 - 74 T + p^{2} T^{2} )^{2}( 1 + 74 T + p^{2} T^{2} )^{2}
67C22C_2^2 (1+6626T2+p4T4)2 ( 1 + 6626 T^{2} + p^{4} T^{4} )^{2}
71C22C_2^2 (1−6482T2+p4T4)2 ( 1 - 6482 T^{2} + p^{4} T^{4} )^{2}
73C2C_2 (1+86T+p2T2)4 ( 1 + 86 T + p^{2} T^{2} )^{4}
79C22C_2^2 (1−11038T2+p4T4)2 ( 1 - 11038 T^{2} + p^{4} T^{4} )^{2}
83C22C_2^2 (1+13586T2+p4T4)2 ( 1 + 13586 T^{2} + p^{4} T^{4} )^{2}
89C2C_2 (1+78T+p2T2)4 ( 1 + 78 T + p^{2} T^{2} )^{4}
97C2C_2 (1−62T+p2T2)4 ( 1 - 62 T + p^{2} 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

−7.38893709189746923848017209327, −7.30303014165155537767067964276, −6.86905918201796305504872498717, −6.86218345965042272201075070055, −6.60871020463681693573581388024, −6.35553075534168364422087835387, −6.24655490795604291462452845787, −5.96676204210569846680043446906, −5.58678463337727409717688986445, −5.07651414132702022736546687496, −5.03844751664607348139678014881, −4.53906099579572994552045647724, −4.52484715522237396919404583613, −4.31880862200366076178991506554, −4.24792991917049444724925928247, −3.81276735275334532379363049809, −3.25247017618835364966359363179, −2.86419562919665502986117135091, −2.69489393388727772336867656866, −2.33530356157797945870074939431, −2.27809206313197820495624340122, −1.62186718045250429957183128699, −1.47183051499872596619843961828, −0.61283480987577486908157251457, −0.12566935799982811304978515457, 0.12566935799982811304978515457, 0.61283480987577486908157251457, 1.47183051499872596619843961828, 1.62186718045250429957183128699, 2.27809206313197820495624340122, 2.33530356157797945870074939431, 2.69489393388727772336867656866, 2.86419562919665502986117135091, 3.25247017618835364966359363179, 3.81276735275334532379363049809, 4.24792991917049444724925928247, 4.31880862200366076178991506554, 4.52484715522237396919404583613, 4.53906099579572994552045647724, 5.03844751664607348139678014881, 5.07651414132702022736546687496, 5.58678463337727409717688986445, 5.96676204210569846680043446906, 6.24655490795604291462452845787, 6.35553075534168364422087835387, 6.60871020463681693573581388024, 6.86218345965042272201075070055, 6.86905918201796305504872498717, 7.30303014165155537767067964276, 7.38893709189746923848017209327

Graph of the ZZ-function along the critical line