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Properties

Label 8-867e4-1.1-c3e4-0-0
Degree 88
Conductor 565036352721565036352721
Sign 11
Analytic cond. 6.84763×1066.84763\times 10^{6}
Root an. cond. 7.152247.15224
Motivic weight 33
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-s − 12·3-s − 8·4-s − 9·5-s − 12·6-s − 35·7-s − 24·8-s + 90·9-s − 9·10-s + 47·11-s + 96·12-s + 85·13-s − 35·14-s + 108·15-s − 3·16-s + 90·18-s − 29·19-s + 72·20-s + 420·21-s + 47·22-s − 60·23-s + 288·24-s − 173·25-s + 85·26-s − 540·27-s + 280·28-s + 39·29-s + ⋯
L(s)  = 1  + 0.353·2-s − 2.30·3-s − 4-s − 0.804·5-s − 0.816·6-s − 1.88·7-s − 1.06·8-s + 10/3·9-s − 0.284·10-s + 1.28·11-s + 2.30·12-s + 1.81·13-s − 0.668·14-s + 1.85·15-s − 0.0468·16-s + 1.17·18-s − 0.350·19-s + 0.804·20-s + 4.36·21-s + 0.455·22-s − 0.543·23-s + 2.44·24-s − 1.38·25-s + 0.641·26-s − 3.84·27-s + 1.88·28-s + 0.249·29-s + ⋯

Functional equation

Λ(s)=((34⋅178)s/2 ΓC(s)4 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((34⋅178)s/2 ΓC(s+3/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 34⋅1783^{4} \cdot 17^{8}
Sign: 11
Analytic conductor: 6.84763×1066.84763\times 10^{6}
Root analytic conductor: 7.152247.15224
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 34⋅178, ( :3/2,3/2,3/2,3/2), 1)(8,\ 3^{4} \cdot 17^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 0.071530340900.07153034090
L(12)L(\frac12) ≈\approx 0.071530340900.07153034090
L(52)L(\frac{5}{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)
bad3C1C_1 (1+pT)4 ( 1 + p T )^{4}
17 1 1
good2C2≀S4C_2 \wr S_4 1−T+9T2+7T3+11p2T4+7p3T5+9p6T6−p9T7+p12T8 1 - T + 9 T^{2} + 7 T^{3} + 11 p^{2} T^{4} + 7 p^{3} T^{5} + 9 p^{6} T^{6} - p^{9} T^{7} + p^{12} T^{8}
5C2≀S4C_2 \wr S_4 1+9T+254T2+423T3+25994T4+423p3T5+254p6T6+9p9T7+p12T8 1 + 9 T + 254 T^{2} + 423 T^{3} + 25994 T^{4} + 423 p^{3} T^{5} + 254 p^{6} T^{6} + 9 p^{9} T^{7} + p^{12} T^{8}
7C2≀S4C_2 \wr S_4 1+5pT+1331T2+29208T3+639932T4+29208p3T5+1331p6T6+5p10T7+p12T8 1 + 5 p T + 1331 T^{2} + 29208 T^{3} + 639932 T^{4} + 29208 p^{3} T^{5} + 1331 p^{6} T^{6} + 5 p^{10} T^{7} + p^{12} T^{8}
11C2≀S4C_2 \wr S_4 1−47T+2608T2−120987T3+5766302T4−120987p3T5+2608p6T6−47p9T7+p12T8 1 - 47 T + 2608 T^{2} - 120987 T^{3} + 5766302 T^{4} - 120987 p^{3} T^{5} + 2608 p^{6} T^{6} - 47 p^{9} T^{7} + p^{12} T^{8}
13C2≀S4C_2 \wr S_4 1−85T+5513T2−132234T3+5981062T4−132234p3T5+5513p6T6−85p9T7+p12T8 1 - 85 T + 5513 T^{2} - 132234 T^{3} + 5981062 T^{4} - 132234 p^{3} T^{5} + 5513 p^{6} T^{6} - 85 p^{9} T^{7} + p^{12} T^{8}
19C2≀S4C_2 \wr S_4 1+29T+13523T2−20244pT3+75083300T4−20244p4T5+13523p6T6+29p9T7+p12T8 1 + 29 T + 13523 T^{2} - 20244 p T^{3} + 75083300 T^{4} - 20244 p^{4} T^{5} + 13523 p^{6} T^{6} + 29 p^{9} T^{7} + p^{12} T^{8}
23C2≀S4C_2 \wr S_4 1+60T+17720T2+1275660T3+382421006T4+1275660p3T5+17720p6T6+60p9T7+p12T8 1 + 60 T + 17720 T^{2} + 1275660 T^{3} + 382421006 T^{4} + 1275660 p^{3} T^{5} + 17720 p^{6} T^{6} + 60 p^{9} T^{7} + p^{12} T^{8}
29C2≀S4C_2 \wr S_4 1−39T+22284T2−174621T3+697436422T4−174621p3T5+22284p6T6−39p9T7+p12T8 1 - 39 T + 22284 T^{2} - 174621 T^{3} + 697436422 T^{4} - 174621 p^{3} T^{5} + 22284 p^{6} T^{6} - 39 p^{9} T^{7} + p^{12} T^{8}
31C2≀S4C_2 \wr S_4 1+342T+76620T2+11555576T3+2166641157T4+11555576p3T5+76620p6T6+342p9T7+p12T8 1 + 342 T + 76620 T^{2} + 11555576 T^{3} + 2166641157 T^{4} + 11555576 p^{3} T^{5} + 76620 p^{6} T^{6} + 342 p^{9} T^{7} + p^{12} T^{8}
37C2≀S4C_2 \wr S_4 1+549T+292215T2+88242794T3+24527184420T4+88242794p3T5+292215p6T6+549p9T7+p12T8 1 + 549 T + 292215 T^{2} + 88242794 T^{3} + 24527184420 T^{4} + 88242794 p^{3} T^{5} + 292215 p^{6} T^{6} + 549 p^{9} T^{7} + p^{12} T^{8}
41C2≀S4C_2 \wr S_4 1−102T+233424T2−16736730T3+22580260702T4−16736730p3T5+233424p6T6−102p9T7+p12T8 1 - 102 T + 233424 T^{2} - 16736730 T^{3} + 22580260702 T^{4} - 16736730 p^{3} T^{5} + 233424 p^{6} T^{6} - 102 p^{9} T^{7} + p^{12} T^{8}
43C2≀S4C_2 \wr S_4 1−797T+411947T2−149795340T3+47337992636T4−149795340p3T5+411947p6T6−797p9T7+p12T8 1 - 797 T + 411947 T^{2} - 149795340 T^{3} + 47337992636 T^{4} - 149795340 p^{3} T^{5} + 411947 p^{6} T^{6} - 797 p^{9} T^{7} + p^{12} T^{8}
47C2≀S4C_2 \wr S_4 1+130T+110812T2−56135910T3−4457775610T4−56135910p3T5+110812p6T6+130p9T7+p12T8 1 + 130 T + 110812 T^{2} - 56135910 T^{3} - 4457775610 T^{4} - 56135910 p^{3} T^{5} + 110812 p^{6} T^{6} + 130 p^{9} T^{7} + p^{12} T^{8}
53C2≀S4C_2 \wr S_4 1+47T+410284T2+2537913T3+81778738982T4+2537913p3T5+410284p6T6+47p9T7+p12T8 1 + 47 T + 410284 T^{2} + 2537913 T^{3} + 81778738982 T^{4} + 2537913 p^{3} T^{5} + 410284 p^{6} T^{6} + 47 p^{9} T^{7} + p^{12} T^{8}
59C2≀S4C_2 \wr S_4 1−549T+876984T2−335147577T3+275814575086T4−335147577p3T5+876984p6T6−549p9T7+p12T8 1 - 549 T + 876984 T^{2} - 335147577 T^{3} + 275814575086 T^{4} - 335147577 p^{3} T^{5} + 876984 p^{6} T^{6} - 549 p^{9} T^{7} + p^{12} T^{8}
61C2≀S4C_2 \wr S_4 1+1175T+1300133T2+847667550T3+489079033138T4+847667550p3T5+1300133p6T6+1175p9T7+p12T8 1 + 1175 T + 1300133 T^{2} + 847667550 T^{3} + 489079033138 T^{4} + 847667550 p^{3} T^{5} + 1300133 p^{6} T^{6} + 1175 p^{9} T^{7} + p^{12} T^{8}
67C2≀S4C_2 \wr S_4 1−1465T+1709677T2−1350002212T3+841967699422T4−1350002212p3T5+1709677p6T6−1465p9T7+p12T8 1 - 1465 T + 1709677 T^{2} - 1350002212 T^{3} + 841967699422 T^{4} - 1350002212 p^{3} T^{5} + 1709677 p^{6} T^{6} - 1465 p^{9} T^{7} + p^{12} T^{8}
71C2≀S4C_2 \wr S_4 1+830T+789084T2+622153702T3+335665885094T4+622153702p3T5+789084p6T6+830p9T7+p12T8 1 + 830 T + 789084 T^{2} + 622153702 T^{3} + 335665885094 T^{4} + 622153702 p^{3} T^{5} + 789084 p^{6} T^{6} + 830 p^{9} T^{7} + p^{12} T^{8}
73C2≀S4C_2 \wr S_4 1−1265T+1890366T2−1457351847T3+1160366421538T4−1457351847p3T5+1890366p6T6−1265p9T7+p12T8 1 - 1265 T + 1890366 T^{2} - 1457351847 T^{3} + 1160366421538 T^{4} - 1457351847 p^{3} T^{5} + 1890366 p^{6} T^{6} - 1265 p^{9} T^{7} + p^{12} T^{8}
79C2≀S4C_2 \wr S_4 1−165T+1327036T2−319061169T3+859091519814T4−319061169p3T5+1327036p6T6−165p9T7+p12T8 1 - 165 T + 1327036 T^{2} - 319061169 T^{3} + 859091519814 T^{4} - 319061169 p^{3} T^{5} + 1327036 p^{6} T^{6} - 165 p^{9} T^{7} + p^{12} T^{8}
83C2≀S4C_2 \wr S_4 1+1814T+1851660T2+949344070T3+548147242838T4+949344070p3T5+1851660p6T6+1814p9T7+p12T8 1 + 1814 T + 1851660 T^{2} + 949344070 T^{3} + 548147242838 T^{4} + 949344070 p^{3} T^{5} + 1851660 p^{6} T^{6} + 1814 p^{9} T^{7} + p^{12} T^{8}
89C2≀S4C_2 \wr S_4 1−1938T+3434456T2−3565166622T3+3612751901102T4−3565166622p3T5+3434456p6T6−1938p9T7+p12T8 1 - 1938 T + 3434456 T^{2} - 3565166622 T^{3} + 3612751901102 T^{4} - 3565166622 p^{3} T^{5} + 3434456 p^{6} T^{6} - 1938 p^{9} T^{7} + p^{12} T^{8}
97C2≀S4C_2 \wr S_4 1+2954T+5863700T2+7891887300T3+8666715233413T4+7891887300p3T5+5863700p6T6+2954p9T7+p12T8 1 + 2954 T + 5863700 T^{2} + 7891887300 T^{3} + 8666715233413 T^{4} + 7891887300 p^{3} T^{5} + 5863700 p^{6} T^{6} + 2954 p^{9} T^{7} + p^{12} 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

−6.80057692709422321435884398261, −6.57129214667124722007299820332, −6.17601899321999053969563088845, −6.16839533160315592464178820179, −6.00491246745158157862317879464, −5.90827047993528931097139748004, −5.44857387613672643547368280549, −5.31043835716853771963338318773, −5.27723636043273141489640138973, −4.49839658129316028247490392548, −4.46428128400703077373382179826, −4.44906657350861140522410876751, −4.01509327835506274070020722466, −3.68533928771034743473322061576, −3.52832361218103052151873446786, −3.50962412419058390715721608632, −3.31881446715921978574992632373, −2.72965653374639346986331205903, −2.01719938353204761671934518986, −2.00452814729683018286633923154, −1.50462515067122332730292076707, −1.05932609296340146512377304434, −0.78055624942054051629420021079, −0.34180945324770858050881548313, −0.10099313425919209192254729945, 0.10099313425919209192254729945, 0.34180945324770858050881548313, 0.78055624942054051629420021079, 1.05932609296340146512377304434, 1.50462515067122332730292076707, 2.00452814729683018286633923154, 2.01719938353204761671934518986, 2.72965653374639346986331205903, 3.31881446715921978574992632373, 3.50962412419058390715721608632, 3.52832361218103052151873446786, 3.68533928771034743473322061576, 4.01509327835506274070020722466, 4.44906657350861140522410876751, 4.46428128400703077373382179826, 4.49839658129316028247490392548, 5.27723636043273141489640138973, 5.31043835716853771963338318773, 5.44857387613672643547368280549, 5.90827047993528931097139748004, 6.00491246745158157862317879464, 6.16839533160315592464178820179, 6.17601899321999053969563088845, 6.57129214667124722007299820332, 6.80057692709422321435884398261

Graph of the ZZ-function along the critical line