L(s) = 1 | − 324·3-s − 1.72e3·5-s + 4.84e3·7-s + 6.56e4·9-s − 1.58e4·11-s − 8.24e4·13-s + 5.59e5·15-s + 1.65e5·17-s − 5.39e5·19-s − 1.56e6·21-s + 7.29e5·23-s − 2.30e6·25-s − 1.06e7·27-s − 1.85e6·29-s + 3.19e6·31-s + 5.12e6·33-s − 8.36e6·35-s − 4.18e6·37-s + 2.67e7·39-s + 2.27e5·41-s − 1.60e6·43-s − 1.13e8·45-s + 1.80e7·47-s − 4.01e7·49-s − 5.37e7·51-s − 2.94e7·53-s + 2.73e7·55-s + ⋯ |
L(s) = 1 | − 2.30·3-s − 1.23·5-s + 0.761·7-s + 10/3·9-s − 0.325·11-s − 0.800·13-s + 2.85·15-s + 0.481·17-s − 0.950·19-s − 1.75·21-s + 0.543·23-s − 1.17·25-s − 3.84·27-s − 0.485·29-s + 0.621·31-s + 0.752·33-s − 0.942·35-s − 0.367·37-s + 1.84·39-s + 0.0125·41-s − 0.0713·43-s − 4.12·45-s + 0.539·47-s − 0.994·49-s − 1.11·51-s − 0.512·53-s + 0.402·55-s + ⋯ |
Λ(s)=(=((228⋅34)s/2ΓC(s)4L(s)Λ(10−s)
Λ(s)=(=((228⋅34)s/2ΓC(s+9/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
228⋅34
|
Sign: |
1
|
Analytic conductor: |
1.52994×109 |
Root analytic conductor: |
14.0632 |
Motivic weight: |
9 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
4
|
Selberg data: |
(8, 228⋅34, ( :9/2,9/2,9/2,9/2), 1)
|
Particular Values
L(5) |
= |
0 |
L(21) |
= |
0 |
L(211) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 3 | C1 | (1+p4T)4 |
good | 5 | C2≀S4 | 1+1728T+5286836T2+1344360512pT3+585667235526p2T4+1344360512p10T5+5286836p18T6+1728p27T7+p36T8 |
| 7 | C2≀S4 | 1−4840T+9079372pT2−9122761384p2T3+9758964239914p3T4−9122761384p11T5+9079372p19T6−4840p27T7+p36T8 |
| 11 | C2≀S4 | 1+15824T+3272030348T2+110832700706512T3+11168899925921994134T4+110832700706512p9T5+3272030348p18T6+15824p27T7+p36T8 |
| 13 | C2≀S4 | 1+82440T+32507843788T2+2806427918898776T3+46⋯74T4+2806427918898776p9T5+32507843788p18T6+82440p27T7+p36T8 |
| 17 | C2≀S4 | 1−165912T+402742640540T2−57573640941296168T3+68⋯90T4−57573640941296168p9T5+402742640540p18T6−165912p27T7+p36T8 |
| 19 | C2≀S4 | 1+28416pT+878148739660T2+411395194240546048T3+35⋯02T4+411395194240546048p9T5+878148739660p18T6+28416p28T7+p36T8 |
| 23 | C2≀S4 | 1−729680T+3870147116348T2−3090843717350364304T3+76⋯54T4−3090843717350364304p9T5+3870147116348p18T6−729680p27T7+p36T8 |
| 29 | C2≀S4 | 1+1850864T+51896421352052T2+76884864756743058576T3+10⋯42T4+76884864756743058576p9T5+51896421352052p18T6+1850864p27T7+p36T8 |
| 31 | C2≀S4 | 1−103160pT+80766069413044T2−19⋯92T3+28⋯74T4−19⋯92p9T5+80766069413044p18T6−103160p28T7+p36T8 |
| 37 | C2≀S4 | 1+4187992T+316603511258092T2+22⋯72T3+48⋯50T4+22⋯72p9T5+316603511258092p18T6+4187992p27T7+p36T8 |
| 41 | C2≀S4 | 1−227704T+815482715263292T2−24⋯80T3+33⋯46T4−24⋯80p9T5+815482715263292p18T6−227704p27T7+p36T8 |
| 43 | C2≀S4 | 1+1600352T+1039425217732396T2−25⋯40T3+54⋯14T4−25⋯40p9T5+1039425217732396p18T6+1600352p27T7+p36T8 |
| 47 | C2≀S4 | 1−18053904T+898850271089564T2−31⋯80T3+25⋯54T4−31⋯80p9T5+898850271089564p18T6−18053904p27T7+p36T8 |
| 53 | C2≀S4 | 1+29418288T+8246475311053268T2+75⋯52T3+31⋯10T4+75⋯52p9T5+8246475311053268p18T6+29418288p27T7+p36T8 |
| 59 | C2≀S4 | 1+38300048T+12480247782213068T2+64⋯60T3+18⋯66T4+64⋯60p9T5+12480247782213068p18T6+38300048p27T7+p36T8 |
| 61 | C2≀S4 | 1−99764648T+35070113945959756T2−25⋯08T3+53⋯54T4−25⋯08p9T5+35070113945959756p18T6−99764648p27T7+p36T8 |
| 67 | C2≀S4 | 1−183717008T+94982362218124012T2−13⋯36T3+56⋯98pT4−13⋯36p9T5+94982362218124012p18T6−183717008p27T7+p36T8 |
| 71 | C2≀S4 | 1−181868080T+187559006549429756T2−24⋯80T3+12⋯86T4−24⋯80p9T5+187559006549429756p18T6−181868080p27T7+p36T8 |
| 73 | C2≀S4 | 1−254539160T+208692556022844412T2−39⋯64T3+17⋯82T4−39⋯64p9T5+208692556022844412p18T6−254539160p27T7+p36T8 |
| 79 | C2≀S4 | 1−831578184T+541079473633225972T2−23⋯64T3+93⋯90T4−23⋯64p9T5+541079473633225972p18T6−831578184p27T7+p36T8 |
| 83 | C2≀S4 | 1−687923952T+569668186782399404T2−18⋯04T3+11⋯50T4−18⋯04p9T5+569668186782399404p18T6−687923952p27T7+p36T8 |
| 89 | C2≀S4 | 1−627627272T+1209510335623979132T2−64⋯44T3+60⋯14T4−64⋯44p9T5+1209510335623979132p18T6−627627272p27T7+p36T8 |
| 97 | C2≀S4 | 1+889385880T+757736193785874268T2+68⋯80T3+12⋯34T4+68⋯80p9T5+757736193785874268p18T6+889385880p27T7+p36T8 |
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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
−7.21953944298647234879238008663, −6.70977829897245237584896270327, −6.56811404743840202616116417650, −6.40366981143595671452677332712, −6.40069119132911164184641280463, −5.84300716679751072964329906050, −5.45509633397555028154597785826, −5.43389612054543715067250299516, −5.29653747536534842203075401286, −4.90532976385749784906739601087, −4.73795703653614007245062783312, −4.45795018179924614251708434492, −4.29563866172337722579707232297, −3.98139426579004466978176649471, −3.57644495740852728200879200734, −3.44447708796586852349358038108, −3.40393120353781581012110011707, −2.56856758471296464469941357079, −2.29251621766701040935145204151, −2.08432114502390631600488445208, −2.07997878355662713525033593925, −1.31408191973900606143495366254, −1.13015958075911418184787062208, −1.04489250663729980680634738429, −0.73616208536664821684434958386, 0, 0, 0, 0,
0.73616208536664821684434958386, 1.04489250663729980680634738429, 1.13015958075911418184787062208, 1.31408191973900606143495366254, 2.07997878355662713525033593925, 2.08432114502390631600488445208, 2.29251621766701040935145204151, 2.56856758471296464469941357079, 3.40393120353781581012110011707, 3.44447708796586852349358038108, 3.57644495740852728200879200734, 3.98139426579004466978176649471, 4.29563866172337722579707232297, 4.45795018179924614251708434492, 4.73795703653614007245062783312, 4.90532976385749784906739601087, 5.29653747536534842203075401286, 5.43389612054543715067250299516, 5.45509633397555028154597785826, 5.84300716679751072964329906050, 6.40069119132911164184641280463, 6.40366981143595671452677332712, 6.56811404743840202616116417650, 6.70977829897245237584896270327, 7.21953944298647234879238008663