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Properties

Label 8-832e4-1.1-c1e4-0-31
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 44

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s + 16·9-s − 4·13-s − 6·17-s − 6·19-s − 6·23-s − 14·25-s − 24·27-s − 12·29-s + 8·37-s + 24·39-s − 18·41-s − 36·43-s − 10·49-s + 36·51-s + 36·57-s − 12·59-s − 36·61-s − 6·67-s + 36·69-s + 30·71-s + 84·75-s − 24·79-s + 21·81-s − 24·83-s + 72·87-s − 12·89-s + ⋯
L(s)  = 1  − 3.46·3-s + 16/3·9-s − 1.10·13-s − 1.45·17-s − 1.37·19-s − 1.25·23-s − 2.79·25-s − 4.61·27-s − 2.22·29-s + 1.31·37-s + 3.84·39-s − 2.81·41-s − 5.48·43-s − 1.42·49-s + 5.04·51-s + 4.76·57-s − 1.56·59-s − 4.60·61-s − 0.733·67-s + 4.33·69-s + 3.56·71-s + 9.69·75-s − 2.70·79-s + 7/3·81-s − 2.63·83-s + 7.71·87-s − 1.27·89-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: 44
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) == 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
13C22C_2^2 1+4T+3T2+4pT3+p2T4 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4}
good3D4×C2D_4\times C_2 1+2pT+20T2+16pT3+91T4+16p2T5+20p2T6+2p4T7+p4T8 1 + 2 p T + 20 T^{2} + 16 p T^{3} + 91 T^{4} + 16 p^{2} T^{5} + 20 p^{2} T^{6} + 2 p^{4} T^{7} + p^{4} T^{8}
5C22C_2^2 (1+7T2+p2T4)2 ( 1 + 7 T^{2} + p^{2} T^{4} )^{2}
7C23C_2^3 1+10T2+51T4+10p2T6+p4T8 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8}
11C23C_2^3 1−10T2−21T4−10p2T6+p4T8 1 - 10 T^{2} - 21 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8}
17D4×C2D_4\times C_2 1+6T+5T2−18T3+60T4−18pT5+5p2T6+6p3T7+p4T8 1 + 6 T + 5 T^{2} - 18 T^{3} + 60 T^{4} - 18 p T^{5} + 5 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
19D4×C2D_4\times C_2 1+6T−8T2+36T3+891T4+36pT5−8p2T6+6p3T7+p4T8 1 + 6 T - 8 T^{2} + 36 T^{3} + 891 T^{4} + 36 p T^{5} - 8 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
23D4×C2D_4\times C_2 1+6T−16T2+36T3+1347T4+36pT5−16p2T6+6p3T7+p4T8 1 + 6 T - 16 T^{2} + 36 T^{3} + 1347 T^{4} + 36 p T^{5} - 16 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
29D4×C2D_4\times C_2 1+12T+109T2+732T3+4272T4+732pT5+109p2T6+12p3T7+p4T8 1 + 12 T + 109 T^{2} + 732 T^{3} + 4272 T^{4} + 732 p T^{5} + 109 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}
31D4×C2D_4\times C_2 1−20T2−678T4−20p2T6+p4T8 1 - 20 T^{2} - 678 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8}
37D4×C2D_4\times C_2 1−8T+T2+88T3+232T4+88pT5+p2T6−8p3T7+p4T8 1 - 8 T + T^{2} + 88 T^{3} + 232 T^{4} + 88 p T^{5} + p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}
41C22C_2^2 (1+9T+68T2+9pT3+p2T4)2 ( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2}
43D4×C2D_4\times C_2 1+36T+622T2+6840T3+52827T4+6840pT5+622p2T6+36p3T7+p4T8 1 + 36 T + 622 T^{2} + 6840 T^{3} + 52827 T^{4} + 6840 p T^{5} + 622 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8}
47D4×C2D_4\times C_2 1−20T2+1818T4−20p2T6+p4T8 1 - 20 T^{2} + 1818 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8}
53D4×C2D_4\times C_2 1−26T2+1899T4−26p2T6+p4T8 1 - 26 T^{2} + 1899 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8}
59D4×C2D_4\times C_2 1+12T+38T2−144T3−741T4−144pT5+38p2T6+12p3T7+p4T8 1 + 12 T + 38 T^{2} - 144 T^{3} - 741 T^{4} - 144 p T^{5} + 38 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}
61D4×C2D_4\times C_2 1+36T+653T2+7956T3+71472T4+7956pT5+653p2T6+36p3T7+p4T8 1 + 36 T + 653 T^{2} + 7956 T^{3} + 71472 T^{4} + 7956 p T^{5} + 653 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8}
67D4×C2D_4\times C_2 1+6T−80T2−108T3+6555T4−108pT5−80p2T6+6p3T7+p4T8 1 + 6 T - 80 T^{2} - 108 T^{3} + 6555 T^{4} - 108 p T^{5} - 80 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
71D4×C2D_4\times C_2 1−30T+508T2−6240T3+59523T4−6240pT5+508p2T6−30p3T7+p4T8 1 - 30 T + 508 T^{2} - 6240 T^{3} + 59523 T^{4} - 6240 p T^{5} + 508 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8}
73C22C_2^2 (1−119T2+p2T4)2 ( 1 - 119 T^{2} + p^{2} T^{4} )^{2}
79C2C_2 (1+6T+pT2)4 ( 1 + 6 T + p T^{2} )^{4}
83D4D_{4} (1+12T+190T2+12pT3+p2T4)2 ( 1 + 12 T + 190 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}
89D4×C2D_4\times C_2 1+12T+94T2+552T3−1533T4+552pT5+94p2T6+12p3T7+p4T8 1 + 12 T + 94 T^{2} + 552 T^{3} - 1533 T^{4} + 552 p T^{5} + 94 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}
97C23C_2^3 1+158T2+15555T4+158p2T6+p4T8 1 + 158 T^{2} + 15555 T^{4} + 158 p^{2} T^{6} + p^{4} 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

−7.72408930626670411408262597742, −7.51688178351082182344109482824, −7.13950416034596584885515816296, −6.83974332914137070798219456170, −6.80090567775694803637690421120, −6.47715012416389482423931812368, −6.43255493432673088387957864134, −6.01685940753180579280486418089, −6.00507102378909348348141838790, −5.72123882800760642411037093895, −5.53510546586307052289615881679, −5.37965047500399563365257126725, −5.09977681012943747586778877994, −4.66932085531096349826805844152, −4.52477829635618333634785034522, −4.42230817266854609772076589345, −4.40493373556656307747180293251, −3.68258941768307417564459961712, −3.42106742100160321591840129485, −3.28261446665017997578343359922, −2.86595080290625693495463666517, −2.03308556137160127111835272593, −1.95926020103333398719218504371, −1.73638610231471740308443801922, −1.55127966613772653635832194470, 0, 0, 0, 0, 1.55127966613772653635832194470, 1.73638610231471740308443801922, 1.95926020103333398719218504371, 2.03308556137160127111835272593, 2.86595080290625693495463666517, 3.28261446665017997578343359922, 3.42106742100160321591840129485, 3.68258941768307417564459961712, 4.40493373556656307747180293251, 4.42230817266854609772076589345, 4.52477829635618333634785034522, 4.66932085531096349826805844152, 5.09977681012943747586778877994, 5.37965047500399563365257126725, 5.53510546586307052289615881679, 5.72123882800760642411037093895, 6.00507102378909348348141838790, 6.01685940753180579280486418089, 6.43255493432673088387957864134, 6.47715012416389482423931812368, 6.80090567775694803637690421120, 6.83974332914137070798219456170, 7.13950416034596584885515816296, 7.51688178351082182344109482824, 7.72408930626670411408262597742

Graph of the ZZ-function along the critical line