↑

Properties

Label 8-24e8-1.1-c2e4-0-12
Degree 88
Conductor 110075314176110075314176
Sign 11
Analytic cond. 60677.860677.8
Root an. cond. 3.961673.96167
Motivic weight 22
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 12·3-s − 6·5-s − 6·7-s + 90·9-s + 18·11-s + 14·13-s − 72·15-s − 8·19-s − 72·21-s + 30·23-s + 3·25-s + 540·27-s − 6·29-s + 74·31-s + 216·33-s + 36·35-s + 120·37-s + 168·39-s − 138·41-s − 10·43-s − 540·45-s + 174·47-s + 11·49-s − 108·55-s − 96·57-s + 18·59-s + 62·61-s + ⋯
L(s)  = 1  + 4·3-s − 6/5·5-s − 6/7·7-s + 10·9-s + 1.63·11-s + 1.07·13-s − 4.79·15-s − 0.421·19-s − 3.42·21-s + 1.30·23-s + 3/25·25-s + 20·27-s − 0.206·29-s + 2.38·31-s + 6.54·33-s + 1.02·35-s + 3.24·37-s + 4.30·39-s − 3.36·41-s − 0.232·43-s − 12·45-s + 3.70·47-s + 0.224·49-s − 1.96·55-s − 1.68·57-s + 0.305·59-s + 1.01·61-s + ⋯

Functional equation

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

Invariants

Degree: 88
Conductor: 224⋅382^{24} \cdot 3^{8}
Sign: 11
Analytic conductor: 60677.860677.8
Root analytic conductor: 3.961673.96167
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 224⋅38, ( :1,1,1,1), 1)(8,\ 2^{24} \cdot 3^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 34.2752053834.27520538
L(12)L(\frac12) ≈\approx 34.2752053834.27520538
L(2)L(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
3C1C_1 (1−pT)4 ( 1 - p T )^{4}
good5D4×C2D_4\times C_2 1+6T+33T2+126T3+116T4+126p2T5+33p4T6+6p6T7+p8T8 1 + 6 T + 33 T^{2} + 126 T^{3} + 116 T^{4} + 126 p^{2} T^{5} + 33 p^{4} T^{6} + 6 p^{6} T^{7} + p^{8} T^{8}
7D4×C2D_4\times C_2 1+6T+25T2−522T3−4044T4−522p2T5+25p4T6+6p6T7+p8T8 1 + 6 T + 25 T^{2} - 522 T^{3} - 4044 T^{4} - 522 p^{2} T^{5} + 25 p^{4} T^{6} + 6 p^{6} T^{7} + p^{8} T^{8}
11D4×C2D_4\times C_2 1−18T+249T2−2538T3+18308T4−2538p2T5+249p4T6−18p6T7+p8T8 1 - 18 T + 249 T^{2} - 2538 T^{3} + 18308 T^{4} - 2538 p^{2} T^{5} + 249 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8}
13D4×C2D_4\times C_2 1−14T−95T2+658T3+22996T4+658p2T5−95p4T6−14p6T7+p8T8 1 - 14 T - 95 T^{2} + 658 T^{3} + 22996 T^{4} + 658 p^{2} T^{5} - 95 p^{4} T^{6} - 14 p^{6} T^{7} + p^{8} T^{8}
17D4×C2D_4\times C_2 1−516T2+135302T4−516p4T6+p8T8 1 - 516 T^{2} + 135302 T^{4} - 516 p^{4} T^{6} + p^{8} T^{8}
19D4D_{4} (1+4T+18pT2+4p2T3+p4T4)2 ( 1 + 4 T + 18 p T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2}
23D4×C2D_4\times C_2 1−30T+1401T2−33030T3+1091060T4−33030p2T5+1401p4T6−30p6T7+p8T8 1 - 30 T + 1401 T^{2} - 33030 T^{3} + 1091060 T^{4} - 33030 p^{2} T^{5} + 1401 p^{4} T^{6} - 30 p^{6} T^{7} + p^{8} T^{8}
29D4×C2D_4\times C_2 1+6T+1409T2+8382T3+1254420T4+8382p2T5+1409p4T6+6p6T7+p8T8 1 + 6 T + 1409 T^{2} + 8382 T^{3} + 1254420 T^{4} + 8382 p^{2} T^{5} + 1409 p^{4} T^{6} + 6 p^{6} T^{7} + p^{8} T^{8}
31D4×C2D_4\times C_2 1−74T+2281T2−94202T3+4022068T4−94202p2T5+2281p4T6−74p6T7+p8T8 1 - 74 T + 2281 T^{2} - 94202 T^{3} + 4022068 T^{4} - 94202 p^{2} T^{5} + 2281 p^{4} T^{6} - 74 p^{6} T^{7} + p^{8} T^{8}
37D4D_{4} (1−60T+3254T2−60p2T3+p4T4)2 ( 1 - 60 T + 3254 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2}
41C22C_2^2 (1+69T+3268T2+69p2T3+p4T4)2 ( 1 + 69 T + 3268 T^{2} + 69 p^{2} T^{3} + p^{4} T^{4} )^{2}
43D4×C2D_4\times C_2 1+10T−2087T2−15110T3+1179268T4−15110p2T5−2087p4T6+10p6T7+p8T8 1 + 10 T - 2087 T^{2} - 15110 T^{3} + 1179268 T^{4} - 15110 p^{2} T^{5} - 2087 p^{4} T^{6} + 10 p^{6} T^{7} + p^{8} T^{8}
47D4×C2D_4\times C_2 1−174T+16745T2−1157622T3+61675956T4−1157622p2T5+16745p4T6−174p6T7+p8T8 1 - 174 T + 16745 T^{2} - 1157622 T^{3} + 61675956 T^{4} - 1157622 p^{2} T^{5} + 16745 p^{4} T^{6} - 174 p^{6} T^{7} + p^{8} T^{8}
53D4×C2D_4\times C_2 1−996T2−9136858T4−996p4T6+p8T8 1 - 996 T^{2} - 9136858 T^{4} - 996 p^{4} T^{6} + p^{8} T^{8}
59D4×C2D_4\times C_2 1−18T+6969T2−123498T3+35331908T4−123498p2T5+6969p4T6−18p6T7+p8T8 1 - 18 T + 6969 T^{2} - 123498 T^{3} + 35331908 T^{4} - 123498 p^{2} T^{5} + 6969 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8}
61D4×C2D_4\times C_2 1−62T−4463T2−53630T3+40856884T4−53630p2T5−4463p4T6−62p6T7+p8T8 1 - 62 T - 4463 T^{2} - 53630 T^{3} + 40856884 T^{4} - 53630 p^{2} T^{5} - 4463 p^{4} T^{6} - 62 p^{6} T^{7} + p^{8} T^{8}
67D4×C2D_4\times C_2 1−22T+985T2+208538T3−22072796T4+208538p2T5+985p4T6−22p6T7+p8T8 1 - 22 T + 985 T^{2} + 208538 T^{3} - 22072796 T^{4} + 208538 p^{2} T^{5} + 985 p^{4} T^{6} - 22 p^{6} T^{7} + p^{8} T^{8}
71D4×C2D_4\times C_2 1−16452T2+117605702T4−16452p4T6+p8T8 1 - 16452 T^{2} + 117605702 T^{4} - 16452 p^{4} T^{6} + p^{8} T^{8}
73D4D_{4} (1−20T+7302T2−20p2T3+p4T4)2 ( 1 - 20 T + 7302 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2}
79D4×C2D_4\times C_2 1+86T−2231T2−245530T3+7570612T4−245530p2T5−2231p4T6+86p6T7+p8T8 1 + 86 T - 2231 T^{2} - 245530 T^{3} + 7570612 T^{4} - 245530 p^{2} T^{5} - 2231 p^{4} T^{6} + 86 p^{6} T^{7} + p^{8} T^{8}
83D4×C2D_4\times C_2 1−66T+9321T2−519354T3+24465668T4−519354p2T5+9321p4T6−66p6T7+p8T8 1 - 66 T + 9321 T^{2} - 519354 T^{3} + 24465668 T^{4} - 519354 p^{2} T^{5} + 9321 p^{4} T^{6} - 66 p^{6} T^{7} + p^{8} T^{8}
89D4×C2D_4\times C_2 1−25924T2+285535302T4−25924p4T6+p8T8 1 - 25924 T^{2} + 285535302 T^{4} - 25924 p^{4} T^{6} + p^{8} T^{8}
97D4×C2D_4\times C_2 1−242T+25489T2−3450194T3+454397668T4−3450194p2T5+25489p4T6−242p6T7+p8T8 1 - 242 T + 25489 T^{2} - 3450194 T^{3} + 454397668 T^{4} - 3450194 p^{2} T^{5} + 25489 p^{4} T^{6} - 242 p^{6} T^{7} + p^{8} T^{8}
show more
show less
   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

−7.67448195319248945577004048149, −7.34482980624141117809351449002, −7.29456403412697544876071671020, −6.88813167569989772737104925355, −6.70494959450269473508175319731, −6.45222081320861973693901882149, −6.26884864496769884922748216417, −6.23273940440639986766501663639, −5.49989184037788732915779209031, −5.05398933174623130114866825982, −4.69977025696506042830821918527, −4.56866169049821793774190876385, −4.26454530284287362355828277931, −3.85027344442195721233138870437, −3.79448366997570974918677648936, −3.70131455162801110440201748004, −3.55177177308958986228403185427, −3.02964242668641056997276204255, −2.60027015351911080305266003717, −2.56718656808557055455405157069, −2.49581172823459236844651768699, −1.71051482389479678916026393753, −1.37819337430813906522954436498, −0.980422110614424418984232618323, −0.837743817370796911723038823222, 0.837743817370796911723038823222, 0.980422110614424418984232618323, 1.37819337430813906522954436498, 1.71051482389479678916026393753, 2.49581172823459236844651768699, 2.56718656808557055455405157069, 2.60027015351911080305266003717, 3.02964242668641056997276204255, 3.55177177308958986228403185427, 3.70131455162801110440201748004, 3.79448366997570974918677648936, 3.85027344442195721233138870437, 4.26454530284287362355828277931, 4.56866169049821793774190876385, 4.69977025696506042830821918527, 5.05398933174623130114866825982, 5.49989184037788732915779209031, 6.23273940440639986766501663639, 6.26884864496769884922748216417, 6.45222081320861973693901882149, 6.70494959450269473508175319731, 6.88813167569989772737104925355, 7.29456403412697544876071671020, 7.34482980624141117809351449002, 7.67448195319248945577004048149

Graph of the ZZ-function along the critical line