L(s) = 1 | + (−0.224 + 0.984i)2-s + (−0.900 + 0.433i)3-s + (0.883 + 0.425i)4-s + (0.222 − 0.974i)5-s + (−0.224 − 0.984i)6-s + (2.01 − 0.970i)7-s + (−1.87 + 2.35i)8-s + (0.623 − 0.781i)9-s + (0.909 + 0.438i)10-s + (3.66 + 4.59i)11-s − 0.980·12-s + (−3.64 − 4.56i)13-s + (0.502 + 2.20i)14-s + (0.222 + 0.974i)15-s + (−0.671 − 0.841i)16-s + 6.31·17-s + ⋯ |
L(s) = 1 | + (−0.158 + 0.695i)2-s + (−0.520 + 0.250i)3-s + (0.441 + 0.212i)4-s + (0.0995 − 0.436i)5-s + (−0.0917 − 0.401i)6-s + (0.761 − 0.366i)7-s + (−0.663 + 0.831i)8-s + (0.207 − 0.260i)9-s + (0.287 + 0.138i)10-s + (1.10 + 1.38i)11-s − 0.283·12-s + (−1.00 − 1.26i)13-s + (0.134 + 0.588i)14-s + (0.0574 + 0.251i)15-s + (−0.167 − 0.210i)16-s + 1.53·17-s + ⋯ |
Λ(s)=(=(435s/2ΓC(s)L(s)(0.165−0.986i)Λ(2−s)
Λ(s)=(=(435s/2ΓC(s+1/2)L(s)(0.165−0.986i)Λ(1−s)
Degree: |
2 |
Conductor: |
435
= 3⋅5⋅29
|
Sign: |
0.165−0.986i
|
Analytic conductor: |
3.47349 |
Root analytic conductor: |
1.86373 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ435(286,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 435, ( :1/2), 0.165−0.986i)
|
Particular Values
L(1) |
≈ |
1.06847+0.903937i |
L(21) |
≈ |
1.06847+0.903937i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(0.900−0.433i)T |
| 5 | 1+(−0.222+0.974i)T |
| 29 | 1+(0.912−5.30i)T |
good | 2 | 1+(0.224−0.984i)T+(−1.80−0.867i)T2 |
| 7 | 1+(−2.01+0.970i)T+(4.36−5.47i)T2 |
| 11 | 1+(−3.66−4.59i)T+(−2.44+10.7i)T2 |
| 13 | 1+(3.64+4.56i)T+(−2.89+12.6i)T2 |
| 17 | 1−6.31T+17T2 |
| 19 | 1+(−1.99−0.962i)T+(11.8+14.8i)T2 |
| 23 | 1+(−0.662−2.90i)T+(−20.7+9.97i)T2 |
| 31 | 1+(0.939−4.11i)T+(−27.9−13.4i)T2 |
| 37 | 1+(−1.49+1.87i)T+(−8.23−36.0i)T2 |
| 41 | 1+9.94T+41T2 |
| 43 | 1+(0.664+2.91i)T+(−38.7+18.6i)T2 |
| 47 | 1+(3.80+4.77i)T+(−10.4+45.8i)T2 |
| 53 | 1+(0.497−2.17i)T+(−47.7−22.9i)T2 |
| 59 | 1−12.6T+59T2 |
| 61 | 1+(−13.6+6.59i)T+(38.0−47.6i)T2 |
| 67 | 1+(−2.55+3.19i)T+(−14.9−65.3i)T2 |
| 71 | 1+(8.21+10.2i)T+(−15.7+69.2i)T2 |
| 73 | 1+(−1.64−7.20i)T+(−65.7+31.6i)T2 |
| 79 | 1+(−1.24+1.56i)T+(−17.5−77.0i)T2 |
| 83 | 1+(0.604+0.290i)T+(51.7+64.8i)T2 |
| 89 | 1+(−2.49+10.9i)T+(−80.1−38.6i)T2 |
| 97 | 1+(9.66+4.65i)T+(60.4+75.8i)T2 |
show more | |
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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
−11.52197164776961858056194013891, −10.28751743127813228737544389797, −9.649481544632378884174635651284, −8.378232336780985908386667713347, −7.47019620270196139444138949483, −6.93812011697575506685543102046, −5.48403038812793549610452048195, −5.00017767219406492849622478246, −3.45917024135931421907112682928, −1.58247020584981938538222267788,
1.17684599147023541277410025023, 2.43457126896889218238745137190, 3.76964362509662216640252626777, 5.31628191514212102709681304468, 6.27950641520725637152006001894, 7.03337961475340549844762589230, 8.248657694135326305709100180236, 9.464974900747157290378780371723, 10.12554266313347727789881028022, 11.40366213291200010909783642089