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Properties

Label 8-7488e4-1.1-c1e4-0-9
Degree 88
Conductor 3.144×10153.144\times 10^{15}
Sign 11
Analytic cond. 1.27812×1071.27812\times 10^{7}
Root an. cond. 7.732527.73252
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 44

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 4·13-s − 14·17-s + 3·25-s − 8·29-s + 2·37-s + 12·41-s + 49-s − 20·53-s − 12·61-s + 8·65-s − 24·73-s + 28·85-s + 24·89-s − 16·97-s + 4·101-s − 30·109-s − 40·113-s − 16·121-s − 14·125-s + 127-s + 131-s + 137-s + 139-s + 16·145-s + 149-s + 151-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.10·13-s − 3.39·17-s + 3/5·25-s − 1.48·29-s + 0.328·37-s + 1.87·41-s + 1/7·49-s − 2.74·53-s − 1.53·61-s + 0.992·65-s − 2.80·73-s + 3.03·85-s + 2.54·89-s − 1.62·97-s + 0.398·101-s − 2.87·109-s − 3.76·113-s − 1.45·121-s − 1.25·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.32·145-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

Λ(s)=((224⋅38⋅134)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((224⋅38⋅134)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 224⋅38⋅1342^{24} \cdot 3^{8} \cdot 13^{4}
Sign: 11
Analytic conductor: 1.27812×1071.27812\times 10^{7}
Root analytic conductor: 7.732527.73252
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 44
Selberg data: (8, 224⋅38⋅134, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{24} \cdot 3^{8} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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
13C1C_1 (1+T)4 ( 1 + T )^{4}
good5D4D_{4} (1+T+pT3+p2T4)2 ( 1 + T + p T^{3} + p^{2} T^{4} )^{2}
7C22≀C2C_2^2 \wr C_2 1−T2+88T4−p2T6+p4T8 1 - T^{2} + 88 T^{4} - p^{2} T^{6} + p^{4} T^{8}
11C22≀C2C_2^2 \wr C_2 1+16T2+142T4+16p2T6+p4T8 1 + 16 T^{2} + 142 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8}
17D4D_{4} (1+7T+36T2+7pT3+p2T4)2 ( 1 + 7 T + 36 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2}
19C22≀C2C_2^2 \wr C_2 1+48T2+1134T4+48p2T6+p4T8 1 + 48 T^{2} + 1134 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8}
23C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
29C2C_2 (1+2T+pT2)4 ( 1 + 2 T + p T^{2} )^{4}
31C22≀C2C_2^2 \wr C_2 1+20T2+1366T4+20p2T6+p4T8 1 + 20 T^{2} + 1366 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8}
37D4D_{4} (1−T+64T2−pT3+p2T4)2 ( 1 - T + 64 T^{2} - p T^{3} + p^{2} T^{4} )^{2}
41D4D_{4} (1−6T+50T2−6pT3+p2T4)2 ( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}
43C22≀C2C_2^2 \wr C_2 1+71T2+4456T4+71p2T6+p4T8 1 + 71 T^{2} + 4456 T^{4} + 71 p^{2} T^{6} + p^{4} T^{8}
47C22≀C2C_2^2 \wr C_2 1+159T2+10728T4+159p2T6+p4T8 1 + 159 T^{2} + 10728 T^{4} + 159 p^{2} T^{6} + p^{4} T^{8}
53D4D_{4} (1+10T+90T2+10pT3+p2T4)2 ( 1 + 10 T + 90 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2}
59C22≀C2C_2^2 \wr C_2 1+48T2−498T4+48p2T6+p4T8 1 + 48 T^{2} - 498 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8}
61D4D_{4} (1+6T+90T2+6pT3+p2T4)2 ( 1 + 6 T + 90 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}
67C22≀C2C_2^2 \wr C_2 1−16T2+8878T4−16p2T6+p4T8 1 - 16 T^{2} + 8878 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8}
71C22≀C2C_2^2 \wr C_2 1+103T2+12232T4+103p2T6+p4T8 1 + 103 T^{2} + 12232 T^{4} + 103 p^{2} T^{6} + p^{4} T^{8}
73C2C_2 (1+6T+pT2)4 ( 1 + 6 T + p T^{2} )^{4}
79C22≀C2C_2^2 \wr C_2 1+92T2+4102T4+92p2T6+p4T8 1 + 92 T^{2} + 4102 T^{4} + 92 p^{2} T^{6} + p^{4} T^{8}
83C22≀C2C_2^2 \wr C_2 1+276T2+32166T4+276p2T6+p4T8 1 + 276 T^{2} + 32166 T^{4} + 276 p^{2} T^{6} + p^{4} T^{8}
89C2C_2 (1−6T+pT2)4 ( 1 - 6 T + p T^{2} )^{4}
97D4D_{4} (1+8T+46T2+8pT3+p2T4)2 ( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} 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

−6.12029391731427315789513145521, −5.62180634999003842183532174611, −5.46833489669855354840731042818, −5.32362483713830418527763329447, −5.24047437668904746894271690896, −4.82108639931415739717508289450, −4.76524156985114923069895045056, −4.52179966955378123593433680420, −4.41581608334191897435239196048, −4.22847109885331744619884987749, −4.17230983459498159093735245994, −3.92214676061834151179740557016, −3.78185648876743735765045551287, −3.22923874769557115535923425146, −3.22828908652212673182644652733, −3.19257279859633192610989461106, −2.71226117087426344504330374653, −2.57667567965592075528854474191, −2.26389928203892770152407244227, −2.24788638689645544235796437667, −2.14373983451169288557061606311, −1.55717328523887547018131238917, −1.45264575210826105889514487468, −1.18695215174687380244086291652, −0.903903083101652986774411356564, 0, 0, 0, 0, 0.903903083101652986774411356564, 1.18695215174687380244086291652, 1.45264575210826105889514487468, 1.55717328523887547018131238917, 2.14373983451169288557061606311, 2.24788638689645544235796437667, 2.26389928203892770152407244227, 2.57667567965592075528854474191, 2.71226117087426344504330374653, 3.19257279859633192610989461106, 3.22828908652212673182644652733, 3.22923874769557115535923425146, 3.78185648876743735765045551287, 3.92214676061834151179740557016, 4.17230983459498159093735245994, 4.22847109885331744619884987749, 4.41581608334191897435239196048, 4.52179966955378123593433680420, 4.76524156985114923069895045056, 4.82108639931415739717508289450, 5.24047437668904746894271690896, 5.32362483713830418527763329447, 5.46833489669855354840731042818, 5.62180634999003842183532174611, 6.12029391731427315789513145521

Graph of the ZZ-function along the critical line