L(s) = 1 | + (−2.23 + 0.133i)5-s + (−1.5 − 2.59i)9-s + 4i·13-s + (3.46 + 2i)17-s + (2 + 3.46i)19-s + (6.92 − 4i)23-s + (4.96 − 0.598i)25-s − 2·29-s + (4 − 6.92i)31-s + (6.92 − 4i)37-s + 6·41-s + 8i·43-s + (3.69 + 5.59i)45-s + (−6.92 + 4i)47-s + (−2 + 3.46i)59-s + ⋯ |
L(s) = 1 | + (−0.998 + 0.0599i)5-s + (−0.5 − 0.866i)9-s + 1.10i·13-s + (0.840 + 0.485i)17-s + (0.458 + 0.794i)19-s + (1.44 − 0.834i)23-s + (0.992 − 0.119i)25-s − 0.371·29-s + (0.718 − 1.24i)31-s + (1.13 − 0.657i)37-s + 0.937·41-s + 1.21i·43-s + (0.550 + 0.834i)45-s + (−1.01 + 0.583i)47-s + (−0.260 + 0.450i)59-s + ⋯ |
Λ(s)=(=(980s/2ΓC(s)L(s)(0.982−0.185i)Λ(2−s)
Λ(s)=(=(980s/2ΓC(s+1/2)L(s)(0.982−0.185i)Λ(1−s)
Degree: |
2 |
Conductor: |
980
= 22⋅5⋅72
|
Sign: |
0.982−0.185i
|
Analytic conductor: |
7.82533 |
Root analytic conductor: |
2.79738 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ980(949,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 980, ( :1/2), 0.982−0.185i)
|
Particular Values
L(1) |
≈ |
1.26846+0.118757i |
L(21) |
≈ |
1.26846+0.118757i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 5 | 1+(2.23−0.133i)T |
| 7 | 1 |
good | 3 | 1+(1.5+2.59i)T2 |
| 11 | 1+(−5.5−9.52i)T2 |
| 13 | 1−4iT−13T2 |
| 17 | 1+(−3.46−2i)T+(8.5+14.7i)T2 |
| 19 | 1+(−2−3.46i)T+(−9.5+16.4i)T2 |
| 23 | 1+(−6.92+4i)T+(11.5−19.9i)T2 |
| 29 | 1+2T+29T2 |
| 31 | 1+(−4+6.92i)T+(−15.5−26.8i)T2 |
| 37 | 1+(−6.92+4i)T+(18.5−32.0i)T2 |
| 41 | 1−6T+41T2 |
| 43 | 1−8iT−43T2 |
| 47 | 1+(6.92−4i)T+(23.5−40.7i)T2 |
| 53 | 1+(26.5+45.8i)T2 |
| 59 | 1+(2−3.46i)T+(−29.5−51.0i)T2 |
| 61 | 1+(−3−5.19i)T+(−30.5+52.8i)T2 |
| 67 | 1+(−6.92−4i)T+(33.5+58.0i)T2 |
| 71 | 1−12T+71T2 |
| 73 | 1+(−3.46−2i)T+(36.5+63.2i)T2 |
| 79 | 1+(2+3.46i)T+(−39.5+68.4i)T2 |
| 83 | 1−83T2 |
| 89 | 1+(5+8.66i)T+(−44.5+77.0i)T2 |
| 97 | 1−12iT−97T2 |
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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
−9.881095382358240547002154553866, −9.199099342813155425546326455796, −8.310126054698841747483603129600, −7.58678863588756123840577605265, −6.63730428256926712623220216119, −5.84734254600871668037574197444, −4.53978395344829857762746738944, −3.77414100600725362662326140358, −2.78372263240156959442536848385, −0.962554427255944045818718986399,
0.845848652453346445710810900749, 2.78359223102205622521061339146, 3.47530894235330696219250094753, 4.96998853271478763571430683095, 5.29699790414913958053695680384, 6.77258980269466721507198350982, 7.64046123905680919581557778955, 8.128474708784298076304312557466, 9.050566794417287782413425083481, 10.03048080107623049725194204300