L(s) = 1 | + (−0.342 − 0.939i)2-s + (0.819 − 0.573i)3-s + (−0.766 + 0.642i)4-s + (0.0871 + 0.996i)5-s + (−0.819 − 0.573i)6-s + (−0.965 − 0.258i)7-s + (0.866 + 0.5i)8-s + (0.342 − 0.939i)9-s + (0.906 − 0.422i)10-s + (−0.258 − 0.965i)11-s + (−0.258 + 0.965i)12-s + (0.173 + 0.984i)13-s + (0.0871 + 0.996i)14-s + (0.642 + 0.766i)15-s + (0.173 − 0.984i)16-s + ⋯ |
L(s) = 1 | + (−0.342 − 0.939i)2-s + (0.819 − 0.573i)3-s + (−0.766 + 0.642i)4-s + (0.0871 + 0.996i)5-s + (−0.819 − 0.573i)6-s + (−0.965 − 0.258i)7-s + (0.866 + 0.5i)8-s + (0.342 − 0.939i)9-s + (0.906 − 0.422i)10-s + (−0.258 − 0.965i)11-s + (−0.258 + 0.965i)12-s + (0.173 + 0.984i)13-s + (0.0871 + 0.996i)14-s + (0.642 + 0.766i)15-s + (0.173 − 0.984i)16-s + ⋯ |
Λ(s)=(=(323s/2ΓR(s+1)L(s)(0.997−0.0753i)Λ(1−s)
Λ(s)=(=(323s/2ΓR(s+1)L(s)(0.997−0.0753i)Λ(1−s)
Degree: |
1 |
Conductor: |
323
= 17⋅19
|
Sign: |
0.997−0.0753i
|
Analytic conductor: |
34.7111 |
Root analytic conductor: |
34.7111 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ323(314,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 323, (1: ), 0.997−0.0753i)
|
Particular Values
L(21) |
≈ |
1.506828502−0.05683926496i |
L(21) |
≈ |
1.506828502−0.05683926496i |
L(1) |
≈ |
0.9493377612−0.3334732555i |
L(1) |
≈ |
0.9493377612−0.3334732555i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 17 | 1 |
| 19 | 1 |
good | 2 | 1+(−0.342−0.939i)T |
| 3 | 1+(0.819−0.573i)T |
| 5 | 1+(0.0871+0.996i)T |
| 7 | 1+(−0.965−0.258i)T |
| 11 | 1+(−0.258−0.965i)T |
| 13 | 1+(0.173+0.984i)T |
| 23 | 1+(−0.0871+0.996i)T |
| 29 | 1+(0.422+0.906i)T |
| 31 | 1+(0.258−0.965i)T |
| 37 | 1+(0.707+0.707i)T |
| 41 | 1+(0.573+0.819i)T |
| 43 | 1+(−0.642+0.766i)T |
| 47 | 1+(0.939+0.342i)T |
| 53 | 1+(0.642+0.766i)T |
| 59 | 1+(0.342+0.939i)T |
| 61 | 1+(0.0871−0.996i)T |
| 67 | 1+(0.939+0.342i)T |
| 71 | 1+(0.0871+0.996i)T |
| 73 | 1+(0.819−0.573i)T |
| 79 | 1+(−0.573−0.819i)T |
| 83 | 1+(−0.866+0.5i)T |
| 89 | 1+(0.173+0.984i)T |
| 97 | 1+(0.906+0.422i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−25.270938483773857166574974679163, −24.37624216092381160662363621231, −23.11896199492205938969758527893, −22.460074728353832516072067237806, −21.271328350167696643547687343274, −20.19056512151142350175262809905, −19.68323459954792705122820783875, −18.58393724817521353199410402987, −17.49712507426034271081959676997, −16.5378243855636843785134194962, −15.73784686027720654810650058380, −15.292941488405112639543601570588, −14.11149061643311918869092332456, −13.1148082405760171512609522753, −12.51480464973733815250919543138, −10.35093655857758582569483644616, −9.81490682307374492273813693314, −8.87696858522090591191337867386, −8.20549035155178899433021003395, −7.12802264732261221374052311851, −5.7683278068157574512104597621, −4.841855382803566039660492070237, −3.85278306952365046957986912304, −2.27931556101184990341272905514, −0.52197493142318869995594297961,
1.070581788946561308561063357628, 2.45932431358397884268827853620, 3.19258908505975175220230531535, 4.00725518470116078726958549904, 6.1132720720620890634409631822, 7.10673568359213952955860137419, 8.08537201976496821554213253998, 9.22314279246485052984449068842, 9.8944095504244044709852088639, 11.02770615454567631269940752687, 11.897587344356576185170486672185, 13.16522762733238226597196336909, 13.6533196137855694132969448539, 14.48334491262920641473465425020, 15.84239360757113224165937012919, 16.97971371620191585084482836710, 18.237847917566086847948091353480, 18.781903290241693894296843492492, 19.396426676683406310410199450581, 20.12559919990118624105625007973, 21.384470681784750658787863527849, 21.86540167062890737203965828021, 23.07075891889905636527923536474, 23.784456341130253679067509789133, 25.25247096670516008109792720430