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Properties

Label 8-3330e4-1.1-c1e4-0-6
Degree 88
Conductor 1.230×10141.230\times 10^{14}
Sign 11
Analytic cond. 499902.499902.
Root an. cond. 5.156565.15656
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 10·4-s − 4·5-s + 4·7-s − 20·8-s + 16·10-s − 2·11-s − 3·13-s − 16·14-s + 35·16-s − 6·17-s + 3·19-s − 40·20-s + 8·22-s + 23-s + 10·25-s + 12·26-s + 40·28-s + 3·29-s + 3·31-s − 56·32-s + 24·34-s − 16·35-s − 4·37-s − 12·38-s + 80·40-s + 11·41-s + ⋯
L(s)  = 1  − 2.82·2-s + 5·4-s − 1.78·5-s + 1.51·7-s − 7.07·8-s + 5.05·10-s − 0.603·11-s − 0.832·13-s − 4.27·14-s + 35/4·16-s − 1.45·17-s + 0.688·19-s − 8.94·20-s + 1.70·22-s + 0.208·23-s + 2·25-s + 2.35·26-s + 7.55·28-s + 0.557·29-s + 0.538·31-s − 9.89·32-s + 4.11·34-s − 2.70·35-s − 0.657·37-s − 1.94·38-s + 12.6·40-s + 1.71·41-s + ⋯

Functional equation

Λ(s)=((24⋅38⋅54⋅374)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((24⋅38⋅54⋅374)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 24⋅38⋅54⋅3742^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4}
Sign: 11
Analytic conductor: 499902.499902.
Root analytic conductor: 5.156565.15656
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 24⋅38⋅54⋅374, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.80278545820.8027854582
L(12)L(\frac12) ≈\approx 0.80278545820.8027854582
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)
bad2C1C_1 (1+T)4 ( 1 + T )^{4}
3 1 1
5C1C_1 (1+T)4 ( 1 + T )^{4}
37C1C_1 (1+T)4 ( 1 + T )^{4}
good7C2≀S4C_2 \wr S_4 1−4T+15T2−44T3+96T4−44pT5+15p2T6−4p3T7+p4T8 1 - 4 T + 15 T^{2} - 44 T^{3} + 96 T^{4} - 44 p T^{5} + 15 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
11C2≀S4C_2 \wr S_4 1+2T+21T2−2T3+180T4−2pT5+21p2T6+2p3T7+p4T8 1 + 2 T + 21 T^{2} - 2 T^{3} + 180 T^{4} - 2 p T^{5} + 21 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
13C2≀S4C_2 \wr S_4 1+3T+14T2+73T3+354T4+73pT5+14p2T6+3p3T7+p4T8 1 + 3 T + 14 T^{2} + 73 T^{3} + 354 T^{4} + 73 p T^{5} + 14 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}
17C2≀S4C_2 \wr S_4 1+6T+21T2−106T3−628T4−106pT5+21p2T6+6p3T7+p4T8 1 + 6 T + 21 T^{2} - 106 T^{3} - 628 T^{4} - 106 p T^{5} + 21 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
19C2≀S4C_2 \wr S_4 1−3T+26T2+37T3+186T4+37pT5+26p2T6−3p3T7+p4T8 1 - 3 T + 26 T^{2} + 37 T^{3} + 186 T^{4} + 37 p T^{5} + 26 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}
23C2≀S4C_2 \wr S_4 1−T+16T2−5T3+958T4−5pT5+16p2T6−p3T7+p4T8 1 - T + 16 T^{2} - 5 T^{3} + 958 T^{4} - 5 p T^{5} + 16 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}
29C2≀S4C_2 \wr S_4 1−3T+34T2−57T3+1498T4−57pT5+34p2T6−3p3T7+p4T8 1 - 3 T + 34 T^{2} - 57 T^{3} + 1498 T^{4} - 57 p T^{5} + 34 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}
31C2≀S4C_2 \wr S_4 1−3T+86T2−9pT3+3442T4−9p2T5+86p2T6−3p3T7+p4T8 1 - 3 T + 86 T^{2} - 9 p T^{3} + 3442 T^{4} - 9 p^{2} T^{5} + 86 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}
41C2≀S4C_2 \wr S_4 1−11T+168T2−1269T3+10334T4−1269pT5+168p2T6−11p3T7+p4T8 1 - 11 T + 168 T^{2} - 1269 T^{3} + 10334 T^{4} - 1269 p T^{5} + 168 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8}
43C2≀S4C_2 \wr S_4 1+5T+96T2+517T3+5838T4+517pT5+96p2T6+5p3T7+p4T8 1 + 5 T + 96 T^{2} + 517 T^{3} + 5838 T^{4} + 517 p T^{5} + 96 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}
47C2≀S4C_2 \wr S_4 1+100T2+96T3+6262T4+96pT5+100p2T6+p4T8 1 + 100 T^{2} + 96 T^{3} + 6262 T^{4} + 96 p T^{5} + 100 p^{2} T^{6} + p^{4} T^{8}
53C2≀S4C_2 \wr S_4 1−22T+333T2−3254T3+27156T4−3254pT5+333p2T6−22p3T7+p4T8 1 - 22 T + 333 T^{2} - 3254 T^{3} + 27156 T^{4} - 3254 p T^{5} + 333 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8}
59C2≀S4C_2 \wr S_4 1−8T+184T2−1096T3+14494T4−1096pT5+184p2T6−8p3T7+p4T8 1 - 8 T + 184 T^{2} - 1096 T^{3} + 14494 T^{4} - 1096 p T^{5} + 184 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}
61C2≀S4C_2 \wr S_4 1+5T+114T2+1111T3+7562T4+1111pT5+114p2T6+5p3T7+p4T8 1 + 5 T + 114 T^{2} + 1111 T^{3} + 7562 T^{4} + 1111 p T^{5} + 114 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}
67C2≀S4C_2 \wr S_4 1−6T+196T2−1174T3+17414T4−1174pT5+196p2T6−6p3T7+p4T8 1 - 6 T + 196 T^{2} - 1174 T^{3} + 17414 T^{4} - 1174 p T^{5} + 196 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
71C2≀S4C_2 \wr S_4 1−18T+228T2−2298T3+22806T4−2298pT5+228p2T6−18p3T7+p4T8 1 - 18 T + 228 T^{2} - 2298 T^{3} + 22806 T^{4} - 2298 p T^{5} + 228 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8}
73C2≀S4C_2 \wr S_4 1+5T+164T2−37T3+150pT4−37pT5+164p2T6+5p3T7+p4T8 1 + 5 T + 164 T^{2} - 37 T^{3} + 150 p T^{4} - 37 p T^{5} + 164 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}
79C2C_2 (1−4T+pT2)4 ( 1 - 4 T + p T^{2} )^{4}
83C2≀S4C_2 \wr S_4 1−T+98T2−953T3+7834T4−953pT5+98p2T6−p3T7+p4T8 1 - T + 98 T^{2} - 953 T^{3} + 7834 T^{4} - 953 p T^{5} + 98 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}
89C2≀S4C_2 \wr S_4 1−5T+122T2+621T3+2666T4+621pT5+122p2T6−5p3T7+p4T8 1 - 5 T + 122 T^{2} + 621 T^{3} + 2666 T^{4} + 621 p T^{5} + 122 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8}
97C2≀S4C_2 \wr S_4 1−7T+152T2−1697T3+20014T4−1697pT5+152p2T6−7p3T7+p4T8 1 - 7 T + 152 T^{2} - 1697 T^{3} + 20014 T^{4} - 1697 p T^{5} + 152 p^{2} T^{6} - 7 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

−6.29410643270562873963651650926, −6.10394814717993867156852841037, −5.71191289886424506655659181555, −5.45296833222026870746612625622, −5.41305324396109122843952938560, −5.07612099289584404579755987640, −4.88844365563967611561423287689, −4.75211608781613119658309285907, −4.50673064796484224749831469050, −4.24082500344435270043478823358, −4.02653694099066377421803355165, −3.72130373609701099854069605820, −3.63064946609788112375752761359, −3.15747580856256755274545797900, −2.97744751594233946518807911358, −2.90803013972299712642272519845, −2.43910836535835245514511280788, −2.22118524646101579447834432215, −2.11943939424732675601966374182, −1.76626570129276185306923217313, −1.70365065409675702201901427759, −0.882877305450239840656309263477, −0.833607448087418395372710935486, −0.69168407326170216395107205197, −0.34693216058570296742577216286, 0.34693216058570296742577216286, 0.69168407326170216395107205197, 0.833607448087418395372710935486, 0.882877305450239840656309263477, 1.70365065409675702201901427759, 1.76626570129276185306923217313, 2.11943939424732675601966374182, 2.22118524646101579447834432215, 2.43910836535835245514511280788, 2.90803013972299712642272519845, 2.97744751594233946518807911358, 3.15747580856256755274545797900, 3.63064946609788112375752761359, 3.72130373609701099854069605820, 4.02653694099066377421803355165, 4.24082500344435270043478823358, 4.50673064796484224749831469050, 4.75211608781613119658309285907, 4.88844365563967611561423287689, 5.07612099289584404579755987640, 5.41305324396109122843952938560, 5.45296833222026870746612625622, 5.71191289886424506655659181555, 6.10394814717993867156852841037, 6.29410643270562873963651650926

Graph of the ZZ-function along the critical line