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Properties

Label 8-24e8-1.1-c1e4-0-11
Degree 88
Conductor 110075314176110075314176
Sign 11
Analytic cond. 447.505447.505
Root an. cond. 2.144612.14461
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  + 4·3-s − 2·5-s + 2·7-s + 6·9-s + 2·11-s + 2·13-s − 8·15-s + 16·19-s + 8·21-s − 6·23-s + 11·25-s − 4·27-s + 10·29-s + 10·31-s + 8·33-s − 4·35-s − 16·37-s + 8·39-s + 14·41-s − 10·43-s − 12·45-s + 2·47-s + 9·49-s − 16·53-s − 4·55-s + 64·57-s − 14·59-s + ⋯
L(s)  = 1  + 2.30·3-s − 0.894·5-s + 0.755·7-s + 2·9-s + 0.603·11-s + 0.554·13-s − 2.06·15-s + 3.67·19-s + 1.74·21-s − 1.25·23-s + 11/5·25-s − 0.769·27-s + 1.85·29-s + 1.79·31-s + 1.39·33-s − 0.676·35-s − 2.63·37-s + 1.28·39-s + 2.18·41-s − 1.52·43-s − 1.78·45-s + 0.291·47-s + 9/7·49-s − 2.19·53-s − 0.539·55-s + 8.47·57-s − 1.82·59-s + ⋯

Functional equation

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

Invariants

Degree: 88
Conductor: 224⋅382^{24} \cdot 3^{8}
Sign: 11
Analytic conductor: 447.505447.505
Root analytic conductor: 2.144612.14461
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 224⋅38, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{24} \cdot 3^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 7.5262433717.526243371
L(12)L(\frac12) ≈\approx 7.5262433717.526243371
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
3C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
good5C22C_2^2 (1+T−4T2+pT3+p2T4)2 ( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} )^{2}
7D4×C2D_4\times C_2 1−2T−5T2+10T3+4T4+10pT5−5p2T6−2p3T7+p4T8 1 - 2 T - 5 T^{2} + 10 T^{3} + 4 T^{4} + 10 p T^{5} - 5 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}
11D4×C2D_4\times C_2 1−2T−13T2+10T3+124T4+10pT5−13p2T6−2p3T7+p4T8 1 - 2 T - 13 T^{2} + 10 T^{3} + 124 T^{4} + 10 p T^{5} - 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}
13D4×C2D_4\times C_2 1−2T+T2+46T3−212T4+46pT5+p2T6−2p3T7+p4T8 1 - 2 T + T^{2} + 46 T^{3} - 212 T^{4} + 46 p T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}
17C22C_2^2 (1+10T2+p2T4)2 ( 1 + 10 T^{2} + p^{2} T^{4} )^{2}
19C2C_2 (1−4T+pT2)4 ( 1 - 4 T + p T^{2} )^{4}
23D4×C2D_4\times C_2 1+6T−13T2+18T3+1044T4+18pT5−13p2T6+6p3T7+p4T8 1 + 6 T - 13 T^{2} + 18 T^{3} + 1044 T^{4} + 18 p T^{5} - 13 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
29D4×C2D_4\times C_2 1−10T+41T2−10T3−260T4−10pT5+41p2T6−10p3T7+p4T8 1 - 10 T + 41 T^{2} - 10 T^{3} - 260 T^{4} - 10 p T^{5} + 41 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}
31D4×C2D_4\times C_2 1−10T+19T2−190T3+2500T4−190pT5+19p2T6−10p3T7+p4T8 1 - 10 T + 19 T^{2} - 190 T^{3} + 2500 T^{4} - 190 p T^{5} + 19 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}
37D4D_{4} (1+8T+66T2+8pT3+p2T4)2 ( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}
41D4×C2D_4\times C_2 1−14T+89T2−350T3+1732T4−350pT5+89p2T6−14p3T7+p4T8 1 - 14 T + 89 T^{2} - 350 T^{3} + 1732 T^{4} - 350 p T^{5} + 89 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8}
43C2C_2×\timesC22C_2^2 (1+10T+pT2)2(1−10T+57T2−10pT3+p2T4) ( 1 + 10 T + p T^{2} )^{2}( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} )
47D4×C2D_4\times C_2 1−2T−37T2+106T3−716T4+106pT5−37p2T6−2p3T7+p4T8 1 - 2 T - 37 T^{2} + 106 T^{3} - 716 T^{4} + 106 p T^{5} - 37 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}
53D4D_{4} (1+8T+98T2+8pT3+p2T4)2 ( 1 + 8 T + 98 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}
59D4×C2D_4\times C_2 1+14T+83T2−70T3−2276T4−70pT5+83p2T6+14p3T7+p4T8 1 + 14 T + 83 T^{2} - 70 T^{3} - 2276 T^{4} - 70 p T^{5} + 83 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8}
61D4×C2D_4\times C_2 1+6T−71T2−90T3+5532T4−90pT5−71p2T6+6p3T7+p4T8 1 + 6 T - 71 T^{2} - 90 T^{3} + 5532 T^{4} - 90 p T^{5} - 71 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
67D4×C2D_4\times C_2 1+10T−5T2−290T3−164T4−290pT5−5p2T6+10p3T7+p4T8 1 + 10 T - 5 T^{2} - 290 T^{3} - 164 T^{4} - 290 p T^{5} - 5 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8}
71D4D_{4} (1−4T+50T2−4pT3+p2T4)2 ( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
73C22C_2^2 (1+122T2+p2T4)2 ( 1 + 122 T^{2} + p^{2} T^{4} )^{2}
79D4×C2D_4\times C_2 1−22T+211T2−2530T3+30052T4−2530pT5+211p2T6−22p3T7+p4T8 1 - 22 T + 211 T^{2} - 2530 T^{3} + 30052 T^{4} - 2530 p T^{5} + 211 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8}
83D4×C2D_4\times C_2 1−6T−133T2−18T3+18684T4−18pT5−133p2T6−6p3T7+p4T8 1 - 6 T - 133 T^{2} - 18 T^{3} + 18684 T^{4} - 18 p T^{5} - 133 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
89D4D_{4} (1+16T+218T2+16pT3+p2T4)2 ( 1 + 16 T + 218 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2}
97D4×C2D_4\times C_2 1+2T−167T2−46T3+19444T4−46pT5−167p2T6+2p3T7+p4T8 1 + 2 T - 167 T^{2} - 46 T^{3} + 19444 T^{4} - 46 p T^{5} - 167 p^{2} T^{6} + 2 p^{3} T^{7} + 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

−7.919990534893391139527431022185, −7.57300857532943257530523585071, −7.41224954358226739939056998737, −7.22137311280688918980388961774, −6.84958210846295113531646742289, −6.80747250635659838157061923861, −6.23983221875339386927075768264, −6.09613362390045556365967160179, −5.86224862027021403070764277253, −5.51521547616245808768252320713, −5.14450479600244695432608722570, −4.85814454422634889427170534330, −4.70538099372624309367350085496, −4.36868652569293850968624610200, −4.29475107872026393678415096385, −3.50842229530506274032217200578, −3.41686170466394717851233553985, −3.39059690869868641099343918450, −3.18698482994409146676985043365, −2.79159711250996208435376553078, −2.48120016549726666044315873454, −2.08323582182060101802531385446, −1.46898640719925528769004951341, −1.30174529726152120009415348975, −0.77996947509189724206880637229, 0.77996947509189724206880637229, 1.30174529726152120009415348975, 1.46898640719925528769004951341, 2.08323582182060101802531385446, 2.48120016549726666044315873454, 2.79159711250996208435376553078, 3.18698482994409146676985043365, 3.39059690869868641099343918450, 3.41686170466394717851233553985, 3.50842229530506274032217200578, 4.29475107872026393678415096385, 4.36868652569293850968624610200, 4.70538099372624309367350085496, 4.85814454422634889427170534330, 5.14450479600244695432608722570, 5.51521547616245808768252320713, 5.86224862027021403070764277253, 6.09613362390045556365967160179, 6.23983221875339386927075768264, 6.80747250635659838157061923861, 6.84958210846295113531646742289, 7.22137311280688918980388961774, 7.41224954358226739939056998737, 7.57300857532943257530523585071, 7.919990534893391139527431022185

Graph of the ZZ-function along the critical line