Properties

Label 8-92e4-1.1-c5e4-0-0
Degree 88
Conductor 7163929671639296
Sign 11
Analytic cond. 47401.647401.6
Root an. cond. 3.841263.84126
Motivic weight 55
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  + 29·3-s + 14·5-s + 248·7-s + 98·9-s − 80·11-s + 1.33e3·13-s + 406·15-s + 1.69e3·17-s + 2.75e3·19-s + 7.19e3·21-s − 2.11e3·23-s − 2.55e3·25-s − 6.14e3·27-s + 9.28e3·29-s + 8.10e3·31-s − 2.32e3·33-s + 3.47e3·35-s + 9.83e3·37-s + 3.85e4·39-s − 1.16e4·41-s + 2.23e4·43-s + 1.37e3·45-s + 2.01e3·47-s − 3.30e3·49-s + 4.90e4·51-s − 5.36e4·53-s − 1.12e3·55-s + ⋯
L(s)  = 1  + 1.86·3-s + 0.250·5-s + 1.91·7-s + 0.403·9-s − 0.199·11-s + 2.18·13-s + 0.465·15-s + 1.41·17-s + 1.74·19-s + 3.55·21-s − 0.834·23-s − 0.817·25-s − 1.62·27-s + 2.04·29-s + 1.51·31-s − 0.370·33-s + 0.479·35-s + 1.18·37-s + 4.06·39-s − 1.08·41-s + 1.84·43-s + 0.101·45-s + 0.132·47-s − 0.196·49-s + 2.63·51-s − 2.62·53-s − 0.0499·55-s + ⋯

Functional equation

Λ(s)=(71639296s/2ΓC(s)4L(s)=(Λ(6s)\begin{aligned}\Lambda(s)=\mathstrut & 71639296 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(71639296s/2ΓC(s+5/2)4L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 71639296 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 88
Conductor: 7163929671639296    =    282342^{8} \cdot 23^{4}
Sign: 11
Analytic conductor: 47401.647401.6
Root analytic conductor: 3.841263.84126
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 71639296, ( :5/2,5/2,5/2,5/2), 1)(8,\ 71639296,\ (\ :5/2, 5/2, 5/2, 5/2),\ 1)

Particular Values

L(3)L(3) \approx 15.4015084815.40150848
L(12)L(\frac12) \approx 15.4015084815.40150848
L(72)L(\frac{7}{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)
bad2 1 1
23C1C_1 (1+p2T)4 ( 1 + p^{2} T )^{4}
good3C2S4C_2 \wr S_4 129T+743T212556T3+25408p2T412556p5T5+743p10T629p15T7+p20T8 1 - 29 T + 743 T^{2} - 12556 T^{3} + 25408 p^{2} T^{4} - 12556 p^{5} T^{5} + 743 p^{10} T^{6} - 29 p^{15} T^{7} + p^{20} T^{8}
5C2S4C_2 \wr S_4 114T+2752T260466T3+20397902T460466p5T5+2752p10T614p15T7+p20T8 1 - 14 T + 2752 T^{2} - 60466 T^{3} + 20397902 T^{4} - 60466 p^{5} T^{5} + 2752 p^{10} T^{6} - 14 p^{15} T^{7} + p^{20} T^{8}
7C2S4C_2 \wr S_4 1248T+64812T211273056T3+1607053606T411273056p5T5+64812p10T6248p15T7+p20T8 1 - 248 T + 64812 T^{2} - 11273056 T^{3} + 1607053606 T^{4} - 11273056 p^{5} T^{5} + 64812 p^{10} T^{6} - 248 p^{15} T^{7} + p^{20} T^{8}
11C2S4C_2 \wr S_4 1+80T+290660T2+55744824T3+53101006470T4+55744824p5T5+290660p10T6+80p15T7+p20T8 1 + 80 T + 290660 T^{2} + 55744824 T^{3} + 53101006470 T^{4} + 55744824 p^{5} T^{5} + 290660 p^{10} T^{6} + 80 p^{15} T^{7} + p^{20} T^{8}
13C2S4C_2 \wr S_4 11331T+1577773T21302902086T3+865880201090T41302902086p5T5+1577773p10T61331p15T7+p20T8 1 - 1331 T + 1577773 T^{2} - 1302902086 T^{3} + 865880201090 T^{4} - 1302902086 p^{5} T^{5} + 1577773 p^{10} T^{6} - 1331 p^{15} T^{7} + p^{20} T^{8}
17C2S4C_2 \wr S_4 11690T+4961508T25513412534T3+9772079939734T45513412534p5T5+4961508p10T61690p15T7+p20T8 1 - 1690 T + 4961508 T^{2} - 5513412534 T^{3} + 9772079939734 T^{4} - 5513412534 p^{5} T^{5} + 4961508 p^{10} T^{6} - 1690 p^{15} T^{7} + p^{20} T^{8}
19C2S4C_2 \wr S_4 12752T+8128308T217616020096T3+29592170455798T417616020096p5T5+8128308p10T62752p15T7+p20T8 1 - 2752 T + 8128308 T^{2} - 17616020096 T^{3} + 29592170455798 T^{4} - 17616020096 p^{5} T^{5} + 8128308 p^{10} T^{6} - 2752 p^{15} T^{7} + p^{20} T^{8}
29C2S4C_2 \wr S_4 19281T+78449853T2495820359354T3+2433205701041866T4495820359354p5T5+78449853p10T69281p15T7+p20T8 1 - 9281 T + 78449853 T^{2} - 495820359354 T^{3} + 2433205701041866 T^{4} - 495820359354 p^{5} T^{5} + 78449853 p^{10} T^{6} - 9281 p^{15} T^{7} + p^{20} T^{8}
31C2S4C_2 \wr S_4 18105T+66178623T2359444338576T3+58105632519184pT4359444338576p5T5+66178623p10T68105p15T7+p20T8 1 - 8105 T + 66178623 T^{2} - 359444338576 T^{3} + 58105632519184 p T^{4} - 359444338576 p^{5} T^{5} + 66178623 p^{10} T^{6} - 8105 p^{15} T^{7} + p^{20} T^{8}
37C2S4C_2 \wr S_4 19834T+230498704T21923723878582T3+22539205501025646T41923723878582p5T5+230498704p10T69834p15T7+p20T8 1 - 9834 T + 230498704 T^{2} - 1923723878582 T^{3} + 22539205501025646 T^{4} - 1923723878582 p^{5} T^{5} + 230498704 p^{10} T^{6} - 9834 p^{15} T^{7} + p^{20} T^{8}
41C2S4C_2 \wr S_4 1+11653T+361294293T2+3478400054238T3+60189745348156162T4+3478400054238p5T5+361294293p10T6+11653p15T7+p20T8 1 + 11653 T + 361294293 T^{2} + 3478400054238 T^{3} + 60189745348156162 T^{4} + 3478400054238 p^{5} T^{5} + 361294293 p^{10} T^{6} + 11653 p^{15} T^{7} + p^{20} T^{8}
43C2S4C_2 \wr S_4 122358T+577397244T27174666378102T3+113874198877296886T47174666378102p5T5+577397244p10T622358p15T7+p20T8 1 - 22358 T + 577397244 T^{2} - 7174666378102 T^{3} + 113874198877296886 T^{4} - 7174666378102 p^{5} T^{5} + 577397244 p^{10} T^{6} - 22358 p^{15} T^{7} + p^{20} T^{8}
47C2S4C_2 \wr S_4 12013T+399140647T2808467096888T3+116385444993555632T4808467096888p5T5+399140647p10T62013p15T7+p20T8 1 - 2013 T + 399140647 T^{2} - 808467096888 T^{3} + 116385444993555632 T^{4} - 808467096888 p^{5} T^{5} + 399140647 p^{10} T^{6} - 2013 p^{15} T^{7} + p^{20} T^{8}
53C2S4C_2 \wr S_4 1+53614T+2439191752T2+68424875378714T3+1687820261377132670T4+68424875378714p5T5+2439191752p10T6+53614p15T7+p20T8 1 + 53614 T + 2439191752 T^{2} + 68424875378714 T^{3} + 1687820261377132670 T^{4} + 68424875378714 p^{5} T^{5} + 2439191752 p^{10} T^{6} + 53614 p^{15} T^{7} + p^{20} T^{8}
59C2S4C_2 \wr S_4 1+19660T+1378013804T2+10463291989260T3+980141794100533590T4+10463291989260p5T5+1378013804p10T6+19660p15T7+p20T8 1 + 19660 T + 1378013804 T^{2} + 10463291989260 T^{3} + 980141794100533590 T^{4} + 10463291989260 p^{5} T^{5} + 1378013804 p^{10} T^{6} + 19660 p^{15} T^{7} + p^{20} T^{8}
61C2S4C_2 \wr S_4 1+43446T+3417849392T2+91494697548354T3+4147050535952226894T4+91494697548354p5T5+3417849392p10T6+43446p15T7+p20T8 1 + 43446 T + 3417849392 T^{2} + 91494697548354 T^{3} + 4147050535952226894 T^{4} + 91494697548354 p^{5} T^{5} + 3417849392 p^{10} T^{6} + 43446 p^{15} T^{7} + p^{20} T^{8}
67C2S4C_2 \wr S_4 1+56544T+2794409764T2+132990518207032T3+5816913041778865638T4+132990518207032p5T5+2794409764p10T6+56544p15T7+p20T8 1 + 56544 T + 2794409764 T^{2} + 132990518207032 T^{3} + 5816913041778865638 T^{4} + 132990518207032 p^{5} T^{5} + 2794409764 p^{10} T^{6} + 56544 p^{15} T^{7} + p^{20} T^{8}
71C2S4C_2 \wr S_4 115921T+6612270215T282213881208992T3+17344052626781522544T482213881208992p5T5+6612270215p10T615921p15T7+p20T8 1 - 15921 T + 6612270215 T^{2} - 82213881208992 T^{3} + 17344052626781522544 T^{4} - 82213881208992 p^{5} T^{5} + 6612270215 p^{10} T^{6} - 15921 p^{15} T^{7} + p^{20} T^{8}
73C2S4C_2 \wr S_4 1+179843T+19592274761T2+1415358504440970T3+75419679497796599486T4+1415358504440970p5T5+19592274761p10T6+179843p15T7+p20T8 1 + 179843 T + 19592274761 T^{2} + 1415358504440970 T^{3} + 75419679497796599486 T^{4} + 1415358504440970 p^{5} T^{5} + 19592274761 p^{10} T^{6} + 179843 p^{15} T^{7} + p^{20} T^{8}
79C2S4C_2 \wr S_4 1116422T+14888621792T21048496876201430T3+71901244815438356894T41048496876201430p5T5+14888621792p10T6116422p15T7+p20T8 1 - 116422 T + 14888621792 T^{2} - 1048496876201430 T^{3} + 71901244815438356894 T^{4} - 1048496876201430 p^{5} T^{5} + 14888621792 p^{10} T^{6} - 116422 p^{15} T^{7} + p^{20} T^{8}
83C2S4C_2 \wr S_4 1+9838T+10143357484T2+85085944797206T3+53318700468816579974T4+85085944797206p5T5+10143357484p10T6+9838p15T7+p20T8 1 + 9838 T + 10143357484 T^{2} + 85085944797206 T^{3} + 53318700468816579974 T^{4} + 85085944797206 p^{5} T^{5} + 10143357484 p^{10} T^{6} + 9838 p^{15} T^{7} + p^{20} T^{8}
89C2S4C_2 \wr S_4 1+1674pT+26163573548T2+27144354291534pT3+ 1 + 1674 p T + 26163573548 T^{2} + 27144354291534 p T^{3} + 22 ⁣ ⁣2222\!\cdots\!22T4+27144354291534p6T5+26163573548p10T6+1674p16T7+p20T8 T^{4} + 27144354291534 p^{6} T^{5} + 26163573548 p^{10} T^{6} + 1674 p^{16} T^{7} + p^{20} T^{8}
97C2S4C_2 \wr S_4 172030T+28610306268T21536383832651522T3+ 1 - 72030 T + 28610306268 T^{2} - 1536383832651522 T^{3} + 34 ⁣ ⁣1434\!\cdots\!14T41536383832651522p5T5+28610306268p10T672030p15T7+p20T8 T^{4} - 1536383832651522 p^{5} T^{5} + 28610306268 p^{10} T^{6} - 72030 p^{15} T^{7} + p^{20} 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

−9.455688567600143406966295226235, −8.919204969976453099477469916975, −8.539285334484289554159995492114, −8.493229421199725669847406581114, −8.333563817592058875759935650141, −7.967037068415288904564592734616, −7.67978816795963632942594731031, −7.61531505706806834743466619665, −7.32093670214553163835965337794, −6.27902560625897239229379272281, −6.26992427684431227669286130683, −6.08762656396737846101137024574, −5.61826898647124550696559629666, −5.21339087819810365207902315072, −4.76995473407866603360595538840, −4.41708796030726615985391047823, −4.17222859905600942509554604027, −3.31287619356490240476198455428, −3.31223888576756110844356334836, −2.79816355862108504622378006332, −2.75515942074681724524511972563, −1.76128974485690617398339249911, −1.41912085942650671140141303815, −1.32484972323579025818879848220, −0.57517370250827340953864886244, 0.57517370250827340953864886244, 1.32484972323579025818879848220, 1.41912085942650671140141303815, 1.76128974485690617398339249911, 2.75515942074681724524511972563, 2.79816355862108504622378006332, 3.31223888576756110844356334836, 3.31287619356490240476198455428, 4.17222859905600942509554604027, 4.41708796030726615985391047823, 4.76995473407866603360595538840, 5.21339087819810365207902315072, 5.61826898647124550696559629666, 6.08762656396737846101137024574, 6.26992427684431227669286130683, 6.27902560625897239229379272281, 7.32093670214553163835965337794, 7.61531505706806834743466619665, 7.67978816795963632942594731031, 7.967037068415288904564592734616, 8.333563817592058875759935650141, 8.493229421199725669847406581114, 8.539285334484289554159995492114, 8.919204969976453099477469916975, 9.455688567600143406966295226235

Graph of the ZZ-function along the critical line