L(s) = 1 | + (−1.85 + 1.85i)2-s + (0.722 − 0.722i)3-s − 2.87i·4-s + (−6.14 + 6.14i)5-s + 2.67i·6-s − 9.45i·7-s + (−2.08 − 2.08i)8-s + 7.95i·9-s − 22.7i·10-s + (3.71 − 3.71i)11-s + (−2.07 − 2.07i)12-s + (0.369 − 0.369i)13-s + (17.5 + 17.5i)14-s + 8.87i·15-s + 19.2·16-s + (−15.3 − 15.3i)17-s + ⋯ |
L(s) = 1 | + (−0.927 + 0.927i)2-s + (0.240 − 0.240i)3-s − 0.718i·4-s + (−1.22 + 1.22i)5-s + 0.446i·6-s − 1.35i·7-s + (−0.260 − 0.260i)8-s + 0.884i·9-s − 2.27i·10-s + (0.337 − 0.337i)11-s + (−0.172 − 0.172i)12-s + (0.0284 − 0.0284i)13-s + (1.25 + 1.25i)14-s + 0.591i·15-s + 1.20·16-s + (−0.905 − 0.905i)17-s + ⋯ |
Λ(s)=(=(197s/2ΓC(s)L(s)(0.516+0.856i)Λ(3−s)
Λ(s)=(=(197s/2ΓC(s+1)L(s)(0.516+0.856i)Λ(1−s)
Degree: |
2 |
Conductor: |
197
|
Sign: |
0.516+0.856i
|
Analytic conductor: |
5.36786 |
Root analytic conductor: |
2.31686 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ197(14,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 197, ( :1), 0.516+0.856i)
|
Particular Values
L(23) |
≈ |
0.285630−0.161297i |
L(21) |
≈ |
0.285630−0.161297i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 197 | 1+(−40.5+192.i)T |
good | 2 | 1+(1.85−1.85i)T−4iT2 |
| 3 | 1+(−0.722+0.722i)T−9iT2 |
| 5 | 1+(6.14−6.14i)T−25iT2 |
| 7 | 1+9.45iT−49T2 |
| 11 | 1+(−3.71+3.71i)T−121iT2 |
| 13 | 1+(−0.369+0.369i)T−169iT2 |
| 17 | 1+(15.3+15.3i)T+289iT2 |
| 19 | 1+17.7iT−361T2 |
| 23 | 1+4.12T+529T2 |
| 29 | 1−10.3T+841T2 |
| 31 | 1+(−34.3+34.3i)T−961iT2 |
| 37 | 1+53.7T+1.36e3T2 |
| 41 | 1−17.7iT−1.68e3T2 |
| 43 | 1+15.8iT−1.84e3T2 |
| 47 | 1+59.0iT−2.20e3T2 |
| 53 | 1+65.5T+2.80e3T2 |
| 59 | 1+1.87T+3.48e3T2 |
| 61 | 1+67.6T+3.72e3T2 |
| 67 | 1+(73.0−73.0i)T−4.48e3iT2 |
| 71 | 1+(77.9+77.9i)T+5.04e3iT2 |
| 73 | 1+(37.2−37.2i)T−5.32e3iT2 |
| 79 | 1+(−4.08−4.08i)T+6.24e3iT2 |
| 83 | 1−53.8iT−6.88e3T2 |
| 89 | 1+(−30.5−30.5i)T+7.92e3iT2 |
| 97 | 1−8.46iT−9.40e3T2 |
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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
−11.77347964627880178866046710315, −10.91894997940126247277011945088, −10.15273518599908420950657806669, −8.709038233024351034502662910885, −7.76087429340537378345163862728, −7.19250898229744500192070462448, −6.59594355002220569552690438059, −4.39045408513033676815479325435, −3.07052435249348439333356355017, −0.25748343455340493497621127713,
1.52185705508447556633305579656, 3.29365206298301854848716064639, 4.59909787161423990173724261785, 6.10772523401405849903843318087, 8.042915757666918741262309139941, 8.826663457179970565891997794897, 9.105452266901446598399253092094, 10.37239624169984081338405931116, 11.66683505820000969862949590038, 12.19274744806313571496429222000