L(s) = 1 | + (−0.982 + 0.982i)2-s + (0.0424 + 0.102i)3-s + 0.0682i·4-s + (−0.142 − 0.0589i)6-s + (1.58 + 0.656i)7-s + (−2.03 − 2.03i)8-s + (2.11 − 2.11i)9-s + (5.35 + 2.21i)11-s + (−0.00698 + 0.00289i)12-s − 1.25·13-s + (−2.20 + 0.912i)14-s + 3.85·16-s + (1.85 + 3.68i)17-s + 4.15i·18-s + (−1.99 − 1.99i)19-s + ⋯ |
L(s) = 1 | + (−0.694 + 0.694i)2-s + (0.0244 + 0.0591i)3-s + 0.0341i·4-s + (−0.0580 − 0.0240i)6-s + (0.599 + 0.248i)7-s + (−0.718 − 0.718i)8-s + (0.704 − 0.704i)9-s + (1.61 + 0.669i)11-s + (−0.00201 + 0.000835i)12-s − 0.347·13-s + (−0.588 + 0.243i)14-s + 0.964·16-s + (0.448 + 0.893i)17-s + 0.978i·18-s + (−0.458 − 0.458i)19-s + ⋯ |
Λ(s)=(=(425s/2ΓC(s)L(s)(0.0448−0.998i)Λ(2−s)
Λ(s)=(=(425s/2ΓC(s+1/2)L(s)(0.0448−0.998i)Λ(1−s)
Degree: |
2 |
Conductor: |
425
= 52⋅17
|
Sign: |
0.0448−0.998i
|
Analytic conductor: |
3.39364 |
Root analytic conductor: |
1.84218 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ425(274,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 425, ( :1/2), 0.0448−0.998i)
|
Particular Values
L(1) |
≈ |
0.797089+0.762107i |
L(21) |
≈ |
0.797089+0.762107i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1 |
| 17 | 1+(−1.85−3.68i)T |
good | 2 | 1+(0.982−0.982i)T−2iT2 |
| 3 | 1+(−0.0424−0.102i)T+(−2.12+2.12i)T2 |
| 7 | 1+(−1.58−0.656i)T+(4.94+4.94i)T2 |
| 11 | 1+(−5.35−2.21i)T+(7.77+7.77i)T2 |
| 13 | 1+1.25T+13T2 |
| 19 | 1+(1.99+1.99i)T+19iT2 |
| 23 | 1+(0.785−1.89i)T+(−16.2−16.2i)T2 |
| 29 | 1+(−1.99−4.80i)T+(−20.5+20.5i)T2 |
| 31 | 1+(2.64−1.09i)T+(21.9−21.9i)T2 |
| 37 | 1+(2.41+5.82i)T+(−26.1+26.1i)T2 |
| 41 | 1+(3.61−8.73i)T+(−28.9−28.9i)T2 |
| 43 | 1+(−5.25−5.25i)T+43iT2 |
| 47 | 1−7.63T+47T2 |
| 53 | 1+(5.09−5.09i)T−53iT2 |
| 59 | 1+(−1.54+1.54i)T−59iT2 |
| 61 | 1+(−2.19+5.30i)T+(−43.1−43.1i)T2 |
| 67 | 1+2.46iT−67T2 |
| 71 | 1+(12.4−5.15i)T+(50.2−50.2i)T2 |
| 73 | 1+(−5.39+2.23i)T+(51.6−51.6i)T2 |
| 79 | 1+(−3.62−1.50i)T+(55.8+55.8i)T2 |
| 83 | 1+(−6.29+6.29i)T−83iT2 |
| 89 | 1+14.3iT−89T2 |
| 97 | 1+(−7.84+3.25i)T+(68.5−68.5i)T2 |
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L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−11.47070401021883123675933044917, −10.16871863943241344725115194064, −9.299489056518308396000233491410, −8.779931267516346143849495517737, −7.65839600630523903854211449907, −6.88382863491993190552044933912, −6.11643551190516452674896671438, −4.50980382676382356940233395076, −3.53759462521202888489081660687, −1.48935913314382377239200469632,
1.07384121665871736601089773931, 2.23241237257769585860035452737, 3.87728965970461800645469058978, 5.08778758650533118669309878371, 6.28753447358901992636463595071, 7.44652506087874088937660780034, 8.458709187530877959450651978232, 9.267178349181861730817730444314, 10.14613457719529235222512777659, 10.83812841809820284586471122661