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Properties

Label 8-832e4-1.1-c1e4-0-22
Degree 88
Conductor 479174066176479174066176
Sign 11
Analytic cond. 1948.051948.05
Root an. cond. 2.577502.57750
Motivic weight 11
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  − 2·3-s − 6·7-s + 5·9-s − 2·11-s + 4·13-s + 6·17-s + 6·19-s + 12·21-s + 6·23-s − 4·25-s − 10·27-s + 6·29-s + 4·33-s + 2·37-s − 8·39-s + 6·41-s − 6·43-s + 24·47-s + 21·49-s − 12·51-s − 12·57-s + 6·59-s + 14·61-s − 30·63-s + 18·67-s − 12·69-s + 6·71-s + ⋯
L(s)  = 1  − 1.15·3-s − 2.26·7-s + 5/3·9-s − 0.603·11-s + 1.10·13-s + 1.45·17-s + 1.37·19-s + 2.61·21-s + 1.25·23-s − 4/5·25-s − 1.92·27-s + 1.11·29-s + 0.696·33-s + 0.328·37-s − 1.28·39-s + 0.937·41-s − 0.914·43-s + 3.50·47-s + 3·49-s − 1.68·51-s − 1.58·57-s + 0.781·59-s + 1.79·61-s − 3.77·63-s + 2.19·67-s − 1.44·69-s + 0.712·71-s + ⋯

Functional equation

Λ(s)=((224⋅134)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((224⋅134)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \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⋅1342^{24} \cdot 13^{4}
Sign: 11
Analytic conductor: 1948.051948.05
Root analytic conductor: 2.577502.57750
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 224⋅134, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{24} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 2.2994021272.299402127
L(12)L(\frac12) ≈\approx 2.2994021272.299402127
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
13C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
good3D4×C2D_4\times C_2 1+2T−T2−2T3+4T4−2pT5−p2T6+2p3T7+p4T8 1 + 2 T - T^{2} - 2 T^{3} + 4 T^{4} - 2 p T^{5} - p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
5C22C_2^2 (1+2T2+p2T4)2 ( 1 + 2 T^{2} + p^{2} T^{4} )^{2}
7D4×C2D_4\times C_2 1+6T+15T2+6pT3+20pT4+6p2T5+15p2T6+6p3T7+p4T8 1 + 6 T + 15 T^{2} + 6 p T^{3} + 20 p T^{4} + 6 p^{2} T^{5} + 15 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
11D4×C2D_4\times C_2 1+2T−T2−34T3−140T4−34pT5−p2T6+2p3T7+p4T8 1 + 2 T - T^{2} - 34 T^{3} - 140 T^{4} - 34 p T^{5} - p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
17D4×C2D_4\times C_2 1−6T+T2−6T3+324T4−6pT5+p2T6−6p3T7+p4T8 1 - 6 T + T^{2} - 6 T^{3} + 324 T^{4} - 6 p T^{5} + p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
19D4×C2D_4\times C_2 1−6T+7T2+54T3−204T4+54pT5+7p2T6−6p3T7+p4T8 1 - 6 T + 7 T^{2} + 54 T^{3} - 204 T^{4} + 54 p T^{5} + 7 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
23D4×C2D_4\times C_2 1−6T−T2+54T3+12T4+54pT5−p2T6−6p3T7+p4T8 1 - 6 T - T^{2} + 54 T^{3} + 12 T^{4} + 54 p T^{5} - p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
29D4×C2D_4\times C_2 1−6T+T2+138T3−660T4+138pT5+p2T6−6p3T7+p4T8 1 - 6 T + T^{2} + 138 T^{3} - 660 T^{4} + 138 p T^{5} + p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
31C22C_2^2 (1+30T2+p2T4)2 ( 1 + 30 T^{2} + p^{2} T^{4} )^{2}
37D4×C2D_4\times C_2 1−2T+T2+142T3−1508T4+142pT5+p2T6−2p3T7+p4T8 1 - 2 T + T^{2} + 142 T^{3} - 1508 T^{4} + 142 p T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}
41D4×C2D_4\times C_2 1−6T−47T2−6T3+3732T4−6pT5−47p2T6−6p3T7+p4T8 1 - 6 T - 47 T^{2} - 6 T^{3} + 3732 T^{4} - 6 p T^{5} - 47 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
43D4×C2D_4\times C_2 1+6T−9T2−246T3−1372T4−246pT5−9p2T6+6p3T7+p4T8 1 + 6 T - 9 T^{2} - 246 T^{3} - 1372 T^{4} - 246 p T^{5} - 9 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
47C2C_2 (1−6T+pT2)4 ( 1 - 6 T + p T^{2} )^{4}
53C22C_2^2 (1+98T2+p2T4)2 ( 1 + 98 T^{2} + p^{2} T^{4} )^{2}
59D4×C2D_4\times C_2 1−6T−73T2+54T3+6276T4+54pT5−73p2T6−6p3T7+p4T8 1 - 6 T - 73 T^{2} + 54 T^{3} + 6276 T^{4} + 54 p T^{5} - 73 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
61C22C_2^2 (1−7T−12T2−7pT3+p2T4)2 ( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2}
67D4×C2D_4\times C_2 1−18T+127T2−1134T3+12612T4−1134pT5+127p2T6−18p3T7+p4T8 1 - 18 T + 127 T^{2} - 1134 T^{3} + 12612 T^{4} - 1134 p T^{5} + 127 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8}
71D4×C2D_4\times C_2 1−6T−97T2+54T3+10092T4+54pT5−97p2T6−6p3T7+p4T8 1 - 6 T - 97 T^{2} + 54 T^{3} + 10092 T^{4} + 54 p T^{5} - 97 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
73D4D_{4} (1−8T+90T2−8pT3+p2T4)2 ( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}
79C2C_2 (1+6T+pT2)4 ( 1 + 6 T + p T^{2} )^{4}
83C2C_2 (1+4T+pT2)4 ( 1 + 4 T + p T^{2} )^{4}
89D4×C2D_4\times C_2 1−18T+97T2−882T3+14772T4−882pT5+97p2T6−18p3T7+p4T8 1 - 18 T + 97 T^{2} - 882 T^{3} + 14772 T^{4} - 882 p T^{5} + 97 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8}
97C22C_2^2 (1−9T−16T2−9pT3+p2T4)2 ( 1 - 9 T - 16 T^{2} - 9 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

−7.42367889985727221066954763027, −7.06463475797698566488590736097, −6.77347482949091301193423200549, −6.69773559035657252024482080805, −6.44605905808942323504532348242, −5.97729111066116453338391042122, −5.88759816220109391128941307254, −5.86287536364522807191495214106, −5.71351726054065319705523389686, −5.09881409939928174720054268498, −5.03117479839358553622332199153, −5.02576607554580027059582804928, −4.48367652267624510098683715905, −3.99108539997565107867849714579, −3.98873509949693845456780606314, −3.54334959693829717513962914932, −3.45918456745962179279567063873, −3.36195251581474341094222528916, −2.74608692525034609031094034858, −2.46564452129749095240690330058, −2.35876940303121592843404105727, −1.66137058791137089865050745018, −1.03454823071485268761463618883, −0.76067898754223866459061005006, −0.70595703498957099556870557272, 0.70595703498957099556870557272, 0.76067898754223866459061005006, 1.03454823071485268761463618883, 1.66137058791137089865050745018, 2.35876940303121592843404105727, 2.46564452129749095240690330058, 2.74608692525034609031094034858, 3.36195251581474341094222528916, 3.45918456745962179279567063873, 3.54334959693829717513962914932, 3.98873509949693845456780606314, 3.99108539997565107867849714579, 4.48367652267624510098683715905, 5.02576607554580027059582804928, 5.03117479839358553622332199153, 5.09881409939928174720054268498, 5.71351726054065319705523389686, 5.86287536364522807191495214106, 5.88759816220109391128941307254, 5.97729111066116453338391042122, 6.44605905808942323504532348242, 6.69773559035657252024482080805, 6.77347482949091301193423200549, 7.06463475797698566488590736097, 7.42367889985727221066954763027

Graph of the ZZ-function along the critical line