L(s) = 1 | + (−0.0784 − 0.996i)3-s + (−0.382 + 0.923i)7-s + (−0.987 + 0.156i)9-s + (0.972 − 0.233i)11-s + (−0.587 + 0.809i)13-s + (0.453 − 0.891i)19-s + (0.951 + 0.309i)21-s + (−0.972 + 0.233i)23-s + (0.233 + 0.972i)27-s + (−0.996 + 0.0784i)29-s + (−0.649 − 0.760i)31-s + (−0.309 − 0.951i)33-s + (0.972 + 0.233i)37-s + (0.852 + 0.522i)39-s + (−0.852 + 0.522i)41-s + ⋯ |
L(s) = 1 | + (−0.0784 − 0.996i)3-s + (−0.382 + 0.923i)7-s + (−0.987 + 0.156i)9-s + (0.972 − 0.233i)11-s + (−0.587 + 0.809i)13-s + (0.453 − 0.891i)19-s + (0.951 + 0.309i)21-s + (−0.972 + 0.233i)23-s + (0.233 + 0.972i)27-s + (−0.996 + 0.0784i)29-s + (−0.649 − 0.760i)31-s + (−0.309 − 0.951i)33-s + (0.972 + 0.233i)37-s + (0.852 + 0.522i)39-s + (−0.852 + 0.522i)41-s + ⋯ |
Λ(s)=(=(1700s/2ΓR(s)L(s)(−0.623+0.781i)Λ(1−s)
Λ(s)=(=(1700s/2ΓR(s)L(s)(−0.623+0.781i)Λ(1−s)
Degree: |
1 |
Conductor: |
1700
= 22⋅52⋅17
|
Sign: |
−0.623+0.781i
|
Analytic conductor: |
7.89476 |
Root analytic conductor: |
7.89476 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1700(1031,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 1700, (0: ), −0.623+0.781i)
|
Particular Values
L(21) |
≈ |
0.1048551655+0.2178954245i |
L(21) |
≈ |
0.1048551655+0.2178954245i |
L(1) |
≈ |
0.7750344450−0.1177339012i |
L(1) |
≈ |
0.7750344450−0.1177339012i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 5 | 1 |
| 17 | 1 |
good | 3 | 1+(−0.0784−0.996i)T |
| 7 | 1+(−0.382+0.923i)T |
| 11 | 1+(0.972−0.233i)T |
| 13 | 1+(−0.587+0.809i)T |
| 19 | 1+(0.453−0.891i)T |
| 23 | 1+(−0.972+0.233i)T |
| 29 | 1+(−0.996+0.0784i)T |
| 31 | 1+(−0.649−0.760i)T |
| 37 | 1+(0.972+0.233i)T |
| 41 | 1+(−0.852+0.522i)T |
| 43 | 1+(−0.707+0.707i)T |
| 47 | 1+(0.951+0.309i)T |
| 53 | 1+(0.891−0.453i)T |
| 59 | 1+(−0.987+0.156i)T |
| 61 | 1+(−0.233−0.972i)T |
| 67 | 1+(0.309+0.951i)T |
| 71 | 1+(−0.0784−0.996i)T |
| 73 | 1+(−0.852−0.522i)T |
| 79 | 1+(−0.649+0.760i)T |
| 83 | 1+(−0.453+0.891i)T |
| 89 | 1+(−0.587−0.809i)T |
| 97 | 1+(−0.996+0.0784i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−20.21732824449432104506578990221, −19.69226451622295386370408234891, −18.56091951991490065723050405925, −17.62201961895278327808480242200, −16.87694125997421713885329751284, −16.54385856126275643629495589270, −15.64226060319783044365655236084, −14.8203605443077300537760982967, −14.26911369881854231690868857044, −13.4990687168156024640627964358, −12.39560873940017563455580637453, −11.79415402995523956013295803417, −10.765834296750277134208072888452, −10.17917939495959482581830766573, −9.627963807385288601582066788867, −8.78638620257080567571544312909, −7.781743726307408088509851454309, −7.00656555530503745410469379654, −5.96042624306117226683205627290, −5.27796275066275678313981842426, −4.11662604504260336042768384704, −3.795886982602081835979313766948, −2.81615508975504654480150001252, −1.47864122577361435312410000328, −0.08619618612333397573105566731,
1.38493900773902843163118548083, 2.1923387761897094386256235958, 3.00867115872285093512379527068, 4.102181295306073192146290363450, 5.27230655932801701422337790773, 6.04586929056525391598076826054, 6.69024825235687523602498859982, 7.45366324052547180653694794512, 8.37204036102776438717951832204, 9.22474291248931553967797859615, 9.65537541496923803921190875981, 11.211473127568846691856417638357, 11.66471577479899614877881213730, 12.25701096132281428938997575130, 13.1014974963335004902351572689, 13.782772951711434407401880981337, 14.60071136718704529915361368352, 15.229128267219606539119849804440, 16.40623680766313733954499425807, 16.85982471328313734766846315810, 17.783669008903321471098641497349, 18.49017574874518273328310699267, 19.00585983830147298372740930063, 19.80868116928908224781212501745, 20.21970572625811836341750565657