L(s) = 1 | + (−0.197 − 0.113i)2-s + (−1.56 + 0.904i)3-s + (−3.97 − 6.88i)4-s + 0.411·6-s + (16.6 − 8.20i)7-s + 3.62i·8-s + (−11.8 + 20.5i)9-s + (−8.85 − 15.3i)11-s + (12.4 + 7.18i)12-s − 62.3i·13-s + (−4.20 − 0.271i)14-s + (−31.3 + 54.3i)16-s + (−75.4 + 43.5i)17-s + (4.67 − 2.69i)18-s + (−50.8 + 88.0i)19-s + ⋯ |
L(s) = 1 | + (−0.0696 − 0.0402i)2-s + (−0.301 + 0.174i)3-s + (−0.496 − 0.860i)4-s + 0.0279·6-s + (0.896 − 0.443i)7-s + 0.160i·8-s + (−0.439 + 0.761i)9-s + (−0.242 − 0.420i)11-s + (0.299 + 0.172i)12-s − 1.32i·13-s + (−0.0802 − 0.00518i)14-s + (−0.490 + 0.849i)16-s + (−1.07 + 0.621i)17-s + (0.0612 − 0.0353i)18-s + (−0.614 + 1.06i)19-s + ⋯ |
Λ(s)=(=(175s/2ΓC(s)L(s)(−0.998−0.0610i)Λ(4−s)
Λ(s)=(=(175s/2ΓC(s+3/2)L(s)(−0.998−0.0610i)Λ(1−s)
Degree: |
2 |
Conductor: |
175
= 52⋅7
|
Sign: |
−0.998−0.0610i
|
Analytic conductor: |
10.3253 |
Root analytic conductor: |
3.21330 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ175(149,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 175, ( :3/2), −0.998−0.0610i)
|
Particular Values
L(2) |
≈ |
0.00831033+0.271791i |
L(21) |
≈ |
0.00831033+0.271791i |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1 |
| 7 | 1+(−16.6+8.20i)T |
good | 2 | 1+(0.197+0.113i)T+(4+6.92i)T2 |
| 3 | 1+(1.56−0.904i)T+(13.5−23.3i)T2 |
| 11 | 1+(8.85+15.3i)T+(−665.5+1.15e3i)T2 |
| 13 | 1+62.3iT−2.19e3T2 |
| 17 | 1+(75.4−43.5i)T+(2.45e3−4.25e3i)T2 |
| 19 | 1+(50.8−88.0i)T+(−3.42e3−5.94e3i)T2 |
| 23 | 1+(81.0+46.8i)T+(6.08e3+1.05e4i)T2 |
| 29 | 1+297.T+2.43e4T2 |
| 31 | 1+(45.6+79.0i)T+(−1.48e4+2.57e4i)T2 |
| 37 | 1+(−243.−140.i)T+(2.53e4+4.38e4i)T2 |
| 41 | 1+271.T+6.89e4T2 |
| 43 | 1+7.81iT−7.95e4T2 |
| 47 | 1+(80.0+46.2i)T+(5.19e4+8.99e4i)T2 |
| 53 | 1+(−156.+90.1i)T+(7.44e4−1.28e5i)T2 |
| 59 | 1+(49.8+86.3i)T+(−1.02e5+1.77e5i)T2 |
| 61 | 1+(217.−376.i)T+(−1.13e5−1.96e5i)T2 |
| 67 | 1+(−399.+230.i)T+(1.50e5−2.60e5i)T2 |
| 71 | 1−518.T+3.57e5T2 |
| 73 | 1+(470.−271.i)T+(1.94e5−3.36e5i)T2 |
| 79 | 1+(−119.+207.i)T+(−2.46e5−4.26e5i)T2 |
| 83 | 1−299.iT−5.71e5T2 |
| 89 | 1+(−527.+913.i)T+(−3.52e5−6.10e5i)T2 |
| 97 | 1+288.iT−9.12e5T2 |
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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
−11.34163163057838496505072318789, −10.72480885895276429498284767172, −10.06449991486838167233863279971, −8.531264938171025100589657001510, −7.893076968487823862382671122412, −6.04117268011339289093362385664, −5.27727765193580942713276935009, −4.12977962733413188062955304589, −1.92409026784255476563088182352, −0.12638435182437419274658111057,
2.22707201202525880712778956549, 4.00692373004389558715164359521, 5.05584871399399048433458150997, 6.60627048329905963964686922626, 7.60849460894682171937023871611, 8.904636113097245755696633767497, 9.286358992909052946453449124354, 11.25223379992738385490454968611, 11.65448070777578219811512262695, 12.67212072192751202940780049356