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 + ⋯ |
Λ(s)=(=(71639296s/2ΓC(s)4L(s)Λ(6−s)
Λ(s)=(=(71639296s/2ΓC(s+5/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
71639296
= 28⋅234
|
Sign: |
1
|
Analytic conductor: |
47401.6 |
Root analytic conductor: |
3.84126 |
Motivic weight: |
5 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 71639296, ( :5/2,5/2,5/2,5/2), 1)
|
Particular Values
L(3) |
≈ |
15.40150848 |
L(21) |
≈ |
15.40150848 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 23 | C1 | (1+p2T)4 |
good | 3 | C2≀S4 | 1−29T+743T2−12556T3+25408p2T4−12556p5T5+743p10T6−29p15T7+p20T8 |
| 5 | C2≀S4 | 1−14T+2752T2−60466T3+20397902T4−60466p5T5+2752p10T6−14p15T7+p20T8 |
| 7 | C2≀S4 | 1−248T+64812T2−11273056T3+1607053606T4−11273056p5T5+64812p10T6−248p15T7+p20T8 |
| 11 | C2≀S4 | 1+80T+290660T2+55744824T3+53101006470T4+55744824p5T5+290660p10T6+80p15T7+p20T8 |
| 13 | C2≀S4 | 1−1331T+1577773T2−1302902086T3+865880201090T4−1302902086p5T5+1577773p10T6−1331p15T7+p20T8 |
| 17 | C2≀S4 | 1−1690T+4961508T2−5513412534T3+9772079939734T4−5513412534p5T5+4961508p10T6−1690p15T7+p20T8 |
| 19 | C2≀S4 | 1−2752T+8128308T2−17616020096T3+29592170455798T4−17616020096p5T5+8128308p10T6−2752p15T7+p20T8 |
| 29 | C2≀S4 | 1−9281T+78449853T2−495820359354T3+2433205701041866T4−495820359354p5T5+78449853p10T6−9281p15T7+p20T8 |
| 31 | C2≀S4 | 1−8105T+66178623T2−359444338576T3+58105632519184pT4−359444338576p5T5+66178623p10T6−8105p15T7+p20T8 |
| 37 | C2≀S4 | 1−9834T+230498704T2−1923723878582T3+22539205501025646T4−1923723878582p5T5+230498704p10T6−9834p15T7+p20T8 |
| 41 | C2≀S4 | 1+11653T+361294293T2+3478400054238T3+60189745348156162T4+3478400054238p5T5+361294293p10T6+11653p15T7+p20T8 |
| 43 | C2≀S4 | 1−22358T+577397244T2−7174666378102T3+113874198877296886T4−7174666378102p5T5+577397244p10T6−22358p15T7+p20T8 |
| 47 | C2≀S4 | 1−2013T+399140647T2−808467096888T3+116385444993555632T4−808467096888p5T5+399140647p10T6−2013p15T7+p20T8 |
| 53 | C2≀S4 | 1+53614T+2439191752T2+68424875378714T3+1687820261377132670T4+68424875378714p5T5+2439191752p10T6+53614p15T7+p20T8 |
| 59 | C2≀S4 | 1+19660T+1378013804T2+10463291989260T3+980141794100533590T4+10463291989260p5T5+1378013804p10T6+19660p15T7+p20T8 |
| 61 | C2≀S4 | 1+43446T+3417849392T2+91494697548354T3+4147050535952226894T4+91494697548354p5T5+3417849392p10T6+43446p15T7+p20T8 |
| 67 | C2≀S4 | 1+56544T+2794409764T2+132990518207032T3+5816913041778865638T4+132990518207032p5T5+2794409764p10T6+56544p15T7+p20T8 |
| 71 | C2≀S4 | 1−15921T+6612270215T2−82213881208992T3+17344052626781522544T4−82213881208992p5T5+6612270215p10T6−15921p15T7+p20T8 |
| 73 | C2≀S4 | 1+179843T+19592274761T2+1415358504440970T3+75419679497796599486T4+1415358504440970p5T5+19592274761p10T6+179843p15T7+p20T8 |
| 79 | C2≀S4 | 1−116422T+14888621792T2−1048496876201430T3+71901244815438356894T4−1048496876201430p5T5+14888621792p10T6−116422p15T7+p20T8 |
| 83 | C2≀S4 | 1+9838T+10143357484T2+85085944797206T3+53318700468816579974T4+85085944797206p5T5+10143357484p10T6+9838p15T7+p20T8 |
| 89 | C2≀S4 | 1+1674pT+26163573548T2+27144354291534pT3+22⋯22T4+27144354291534p6T5+26163573548p10T6+1674p16T7+p20T8 |
| 97 | C2≀S4 | 1−72030T+28610306268T2−1536383832651522T3+34⋯14T4−1536383832651522p5T5+28610306268p10T6−72030p15T7+p20T8 |
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L(s)=p∏ j=1∏8(1−αj,pp−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