Properties

Label 8-315e4-1.1-c9e4-0-5
Degree 88
Conductor 98456006259845600625
Sign 11
Analytic cond. 6.92774×1086.92774\times 10^{8}
Root an. cond. 12.737212.7372
Motivic weight 99
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 44

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·2-s − 537·4-s + 2.50e3·5-s − 9.60e3·7-s + 991·8-s − 1.25e4·10-s − 6.45e4·11-s − 2.93e4·13-s + 4.80e4·14-s + 1.69e5·16-s − 2.78e5·17-s − 9.29e5·19-s − 1.34e6·20-s + 3.22e5·22-s + 5.26e5·23-s + 3.90e6·25-s + 1.46e5·26-s + 5.15e6·28-s − 3.65e5·29-s − 4.97e6·31-s − 5.47e5·32-s + 1.39e6·34-s − 2.40e7·35-s + 2.28e7·37-s + 4.64e6·38-s + 2.47e6·40-s + 2.82e7·41-s + ⋯
L(s)  = 1  − 0.220·2-s − 1.04·4-s + 1.78·5-s − 1.51·7-s + 0.0855·8-s − 0.395·10-s − 1.32·11-s − 0.285·13-s + 0.334·14-s + 0.648·16-s − 0.809·17-s − 1.63·19-s − 1.87·20-s + 0.293·22-s + 0.391·23-s + 2·25-s + 0.0630·26-s + 1.58·28-s − 0.0960·29-s − 0.968·31-s − 0.0922·32-s + 0.178·34-s − 2.70·35-s + 2.00·37-s + 0.361·38-s + 0.153·40-s + 1.56·41-s + ⋯

Functional equation

Λ(s)=((385474)s/2ΓC(s)4L(s)=(Λ(10s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}
Λ(s)=((385474)s/2ΓC(s+9/2)4L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 3854743^{8} \cdot 5^{4} \cdot 7^{4}
Sign: 11
Analytic conductor: 6.92774×1086.92774\times 10^{8}
Root analytic conductor: 12.737212.7372
Motivic weight: 99
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 44
Selberg data: (8, 385474, ( :9/2,9/2,9/2,9/2), 1)(8,\ 3^{8} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )

Particular Values

L(5)L(5) == 00
L(12)L(\frac12) == 00
L(112)L(\frac{11}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad3 1 1
5C1C_1 (1p4T)4 ( 1 - p^{4} T )^{4}
7C1C_1 (1+p4T)4 ( 1 + p^{4} T )^{4}
good2C2S4C_2 \wr S_4 1+5T+281pT2+563p3T3+1167p7T4+563p12T5+281p19T6+5p27T7+p36T8 1 + 5 T + 281 p T^{2} + 563 p^{3} T^{3} + 1167 p^{7} T^{4} + 563 p^{12} T^{5} + 281 p^{19} T^{6} + 5 p^{27} T^{7} + p^{36} T^{8}
11C2S4C_2 \wr S_4 1+64546T+7269451996T2+408023915576066T3+24310519363699903350T4+408023915576066p9T5+7269451996p18T6+64546p27T7+p36T8 1 + 64546 T + 7269451996 T^{2} + 408023915576066 T^{3} + 24310519363699903350 T^{4} + 408023915576066 p^{9} T^{5} + 7269451996 p^{18} T^{6} + 64546 p^{27} T^{7} + p^{36} T^{8}
13C2S4C_2 \wr S_4 1+29390T+22351274472T2+265168323158506T3+ 1 + 29390 T + 22351274472 T^{2} + 265168323158506 T^{3} + 25 ⁣ ⁣6225\!\cdots\!62T4+265168323158506p9T5+22351274472p18T6+29390p27T7+p36T8 T^{4} + 265168323158506 p^{9} T^{5} + 22351274472 p^{18} T^{6} + 29390 p^{27} T^{7} + p^{36} T^{8}
17C2S4C_2 \wr S_4 1+278788T+473329598692T2+95909056587305276T3+ 1 + 278788 T + 473329598692 T^{2} + 95909056587305276 T^{3} + 49 ⁣ ⁣9449\!\cdots\!94pT4+95909056587305276p9T5+473329598692p18T6+278788p27T7+p36T8 p T^{4} + 95909056587305276 p^{9} T^{5} + 473329598692 p^{18} T^{6} + 278788 p^{27} T^{7} + p^{36} T^{8}
19C2S4C_2 \wr S_4 1+929142T+965216188644T2+622557929988811782T3+ 1 + 929142 T + 965216188644 T^{2} + 622557929988811782 T^{3} + 42 ⁣ ⁣3042\!\cdots\!30T4+622557929988811782p9T5+965216188644p18T6+929142p27T7+p36T8 T^{4} + 622557929988811782 p^{9} T^{5} + 965216188644 p^{18} T^{6} + 929142 p^{27} T^{7} + p^{36} T^{8}
23C2S4C_2 \wr S_4 1526064T+2170511578012T21872889874671841968T3+ 1 - 526064 T + 2170511578012 T^{2} - 1872889874671841968 T^{3} + 72 ⁣ ⁣6272\!\cdots\!62T41872889874671841968p9T5+2170511578012p18T6526064p27T7+p36T8 T^{4} - 1872889874671841968 p^{9} T^{5} + 2170511578012 p^{18} T^{6} - 526064 p^{27} T^{7} + p^{36} T^{8}
29C2S4C_2 \wr S_4 1+365860T+37670299711220T2+31056883269757829164T3+ 1 + 365860 T + 37670299711220 T^{2} + 31056883269757829164 T^{3} + 67 ⁣ ⁣9867\!\cdots\!98T4+31056883269757829164p9T5+37670299711220p18T6+365860p27T7+p36T8 T^{4} + 31056883269757829164 p^{9} T^{5} + 37670299711220 p^{18} T^{6} + 365860 p^{27} T^{7} + p^{36} T^{8}
31C2S4C_2 \wr S_4 1+4977954T+88072246168420T2+ 1 + 4977954 T + 88072246168420 T^{2} + 37 ⁣ ⁣3437\!\cdots\!34T3+ T^{3} + 32 ⁣ ⁣6232\!\cdots\!62T4+ T^{4} + 37 ⁣ ⁣3437\!\cdots\!34p9T5+88072246168420p18T6+4977954p27T7+p36T8 p^{9} T^{5} + 88072246168420 p^{18} T^{6} + 4977954 p^{27} T^{7} + p^{36} T^{8}
37C2S4C_2 \wr S_4 122846008T+605448015384460T2 1 - 22846008 T + 605448015384460 T^{2} - 79 ⁣ ⁣3279\!\cdots\!32T3+ T^{3} + 11 ⁣ ⁣9011\!\cdots\!90T4 T^{4} - 79 ⁣ ⁣3279\!\cdots\!32p9T5+605448015384460p18T622846008p27T7+p36T8 p^{9} T^{5} + 605448015384460 p^{18} T^{6} - 22846008 p^{27} T^{7} + p^{36} T^{8}
41C2S4C_2 \wr S_4 128257844T+1472108304294052T2 1 - 28257844 T + 1472108304294052 T^{2} - 27 ⁣ ⁣3227\!\cdots\!32T3+ T^{3} + 74 ⁣ ⁣0274\!\cdots\!02T4 T^{4} - 27 ⁣ ⁣3227\!\cdots\!32p9T5+1472108304294052p18T628257844p27T7+p36T8 p^{9} T^{5} + 1472108304294052 p^{18} T^{6} - 28257844 p^{27} T^{7} + p^{36} T^{8}
43C2S4C_2 \wr S_4 123603420T+1942778777119212T2 1 - 23603420 T + 1942778777119212 T^{2} - 34 ⁣ ⁣6434\!\cdots\!64T3+ T^{3} + 14 ⁣ ⁣2214\!\cdots\!22T4 T^{4} - 34 ⁣ ⁣6434\!\cdots\!64p9T5+1942778777119212p18T623603420p27T7+p36T8 p^{9} T^{5} + 1942778777119212 p^{18} T^{6} - 23603420 p^{27} T^{7} + p^{36} T^{8}
47C2S4C_2 \wr S_4 130058700T+3867120056326412T2 1 - 30058700 T + 3867120056326412 T^{2} - 90 ⁣ ⁣7290\!\cdots\!72T3+ T^{3} + 62 ⁣ ⁣9462\!\cdots\!94T4 T^{4} - 90 ⁣ ⁣7290\!\cdots\!72p9T5+3867120056326412p18T630058700p27T7+p36T8 p^{9} T^{5} + 3867120056326412 p^{18} T^{6} - 30058700 p^{27} T^{7} + p^{36} T^{8}
53C2S4C_2 \wr S_4 1+113767294T+14444775155815384T2+ 1 + 113767294 T + 14444775155815384 T^{2} + 90 ⁣ ⁣3090\!\cdots\!30T3+ T^{3} + 67 ⁣ ⁣4667\!\cdots\!46T4+ T^{4} + 90 ⁣ ⁣3090\!\cdots\!30p9T5+14444775155815384p18T6+113767294p27T7+p36T8 p^{9} T^{5} + 14444775155815384 p^{18} T^{6} + 113767294 p^{27} T^{7} + p^{36} T^{8}
59C2S4C_2 \wr S_4 1151786220T+26874891828787196T2 1 - 151786220 T + 26874891828787196 T^{2} - 32 ⁣ ⁣5632\!\cdots\!56T3+ T^{3} + 34 ⁣ ⁣4634\!\cdots\!46T4 T^{4} - 32 ⁣ ⁣5632\!\cdots\!56p9T5+26874891828787196p18T6151786220p27T7+p36T8 p^{9} T^{5} + 26874891828787196 p^{18} T^{6} - 151786220 p^{27} T^{7} + p^{36} T^{8}
61C2S4C_2 \wr S_4 1+191130108T+35831813595938644T2+ 1 + 191130108 T + 35831813595938644 T^{2} + 51 ⁣ ⁣3251\!\cdots\!32T3+ T^{3} + 54 ⁣ ⁣3054\!\cdots\!30T4+ T^{4} + 51 ⁣ ⁣3251\!\cdots\!32p9T5+35831813595938644p18T6+191130108p27T7+p36T8 p^{9} T^{5} + 35831813595938644 p^{18} T^{6} + 191130108 p^{27} T^{7} + p^{36} T^{8}
67C2S4C_2 \wr S_4 1+147812356T+86585970985974924T2+ 1 + 147812356 T + 86585970985974924 T^{2} + 10 ⁣ ⁣1610\!\cdots\!16T3+ T^{3} + 33 ⁣ ⁣5033\!\cdots\!50T4+ T^{4} + 10 ⁣ ⁣1610\!\cdots\!16p9T5+86585970985974924p18T6+147812356p27T7+p36T8 p^{9} T^{5} + 86585970985974924 p^{18} T^{6} + 147812356 p^{27} T^{7} + p^{36} T^{8}
71C2S4C_2 \wr S_4 137100486T+169585368015908684T2 1 - 37100486 T + 169585368015908684 T^{2} - 56 ⁣ ⁣9456\!\cdots\!94T3+ T^{3} + 15 ⁣ ⁣5015\!\cdots\!50pT4 p T^{4} - 56 ⁣ ⁣9456\!\cdots\!94p9T5+169585368015908684p18T637100486p27T7+p36T8 p^{9} T^{5} + 169585368015908684 p^{18} T^{6} - 37100486 p^{27} T^{7} + p^{36} T^{8}
73C2S4C_2 \wr S_4 1163444574T+198269192296166088T2 1 - 163444574 T + 198269192296166088 T^{2} - 25 ⁣ ⁣0625\!\cdots\!06T3+ T^{3} + 16 ⁣ ⁣6216\!\cdots\!62T4 T^{4} - 25 ⁣ ⁣0625\!\cdots\!06p9T5+198269192296166088p18T6163444574p27T7+p36T8 p^{9} T^{5} + 198269192296166088 p^{18} T^{6} - 163444574 p^{27} T^{7} + p^{36} T^{8}
79C2S4C_2 \wr S_4 1+327433200T+359528071713938556T2+ 1 + 327433200 T + 359528071713938556 T^{2} + 94 ⁣ ⁣0894\!\cdots\!08T3+ T^{3} + 62 ⁣ ⁣8662\!\cdots\!86T4+ T^{4} + 94 ⁣ ⁣0894\!\cdots\!08p9T5+359528071713938556p18T6+327433200p27T7+p36T8 p^{9} T^{5} + 359528071713938556 p^{18} T^{6} + 327433200 p^{27} T^{7} + p^{36} T^{8}
83C2S4C_2 \wr S_4 1+216775352T+15468188990873420T2+ 1 + 216775352 T + 15468188990873420 T^{2} + 41 ⁣ ⁣1641\!\cdots\!16T3+ T^{3} + 73 ⁣ ⁣9873\!\cdots\!98T4+ T^{4} + 41 ⁣ ⁣1641\!\cdots\!16p9T5+15468188990873420p18T6+216775352p27T7+p36T8 p^{9} T^{5} + 15468188990873420 p^{18} T^{6} + 216775352 p^{27} T^{7} + p^{36} T^{8}
89C2S4C_2 \wr S_4 1792987912T+1284349731754908604T2 1 - 792987912 T + 1284349731754908604 T^{2} - 68 ⁣ ⁣3668\!\cdots\!36T3+ T^{3} + 64 ⁣ ⁣9064\!\cdots\!90T4 T^{4} - 68 ⁣ ⁣3668\!\cdots\!36p9T5+1284349731754908604p18T6792987912p27T7+p36T8 p^{9} T^{5} + 1284349731754908604 p^{18} T^{6} - 792987912 p^{27} T^{7} + p^{36} T^{8}
97C2S4C_2 \wr S_4 1+640730334T+1921696179196466736T2+ 1 + 640730334 T + 1921696179196466736 T^{2} + 80 ⁣ ⁣7880\!\cdots\!78T3+ T^{3} + 18 ⁣ ⁣9818\!\cdots\!98T4+ T^{4} + 80 ⁣ ⁣7880\!\cdots\!78p9T5+1921696179196466736p18T6+640730334p27T7+p36T8 p^{9} T^{5} + 1921696179196466736 p^{18} T^{6} + 640730334 p^{27} T^{7} + p^{36} T^{8}
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   L(s)=p j=18(1αj,pps)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

−7.32286952252166934542356455881, −7.19455851909712716526031228806, −6.78538779704046950693374848133, −6.53272801875708598740682084733, −6.22881664407345681420619445058, −6.10960683413165327726538418903, −5.83007043394838359157805641403, −5.66810737170310285334505618964, −5.64228315898845193410868676009, −4.87290866606344729493325032800, −4.78215578802665757611940850140, −4.65058392081981738471723867619, −4.51572843527356287686198526505, −3.84105730711281393251247700434, −3.71970605952549733346946210914, −3.57487286213784815634382578955, −2.96229214630390096444131852386, −2.66836263546744812821894502860, −2.57363592073880507451737530294, −2.34714115200587533640154134117, −2.16340944345734252717458307741, −1.76216374968790785481751072595, −1.13954627883648156359915491816, −1.00762893299744498564416093512, −0.959118277880721762417210956129, 0, 0, 0, 0, 0.959118277880721762417210956129, 1.00762893299744498564416093512, 1.13954627883648156359915491816, 1.76216374968790785481751072595, 2.16340944345734252717458307741, 2.34714115200587533640154134117, 2.57363592073880507451737530294, 2.66836263546744812821894502860, 2.96229214630390096444131852386, 3.57487286213784815634382578955, 3.71970605952549733346946210914, 3.84105730711281393251247700434, 4.51572843527356287686198526505, 4.65058392081981738471723867619, 4.78215578802665757611940850140, 4.87290866606344729493325032800, 5.64228315898845193410868676009, 5.66810737170310285334505618964, 5.83007043394838359157805641403, 6.10960683413165327726538418903, 6.22881664407345681420619445058, 6.53272801875708598740682084733, 6.78538779704046950693374848133, 7.19455851909712716526031228806, 7.32286952252166934542356455881

Graph of the ZZ-function along the critical line