L(s) = 1 | + (1.08 − 0.912i)2-s + (1.34 − 1.34i)3-s + (0.335 − 1.97i)4-s + (−2.21 + 0.272i)5-s + (0.225 − 2.67i)6-s + (−0.697 − 0.697i)7-s + (−1.43 − 2.43i)8-s − 0.592i·9-s + (−2.14 + 2.31i)10-s − 4.74i·11-s + (−2.19 − 3.09i)12-s + (3.30 + 3.30i)13-s + (−1.39 − 0.117i)14-s + (−2.60 + 3.33i)15-s + (−3.77 − 1.32i)16-s + (−1.03 + 1.03i)17-s + ⋯ |
L(s) = 1 | + (0.764 − 0.645i)2-s + (0.773 − 0.773i)3-s + (0.167 − 0.985i)4-s + (−0.992 + 0.121i)5-s + (0.0920 − 1.09i)6-s + (−0.263 − 0.263i)7-s + (−0.507 − 0.861i)8-s − 0.197i·9-s + (−0.679 + 0.733i)10-s − 1.43i·11-s + (−0.633 − 0.892i)12-s + (0.916 + 0.916i)13-s + (−0.371 − 0.0313i)14-s + (−0.673 + 0.862i)15-s + (−0.943 − 0.330i)16-s + (−0.249 + 0.249i)17-s + ⋯ |
Λ(s)=(=(380s/2ΓC(s)L(s)(−0.564+0.825i)Λ(2−s)
Λ(s)=(=(380s/2ΓC(s+1/2)L(s)(−0.564+0.825i)Λ(1−s)
Degree: |
2 |
Conductor: |
380
= 22⋅5⋅19
|
Sign: |
−0.564+0.825i
|
Analytic conductor: |
3.03431 |
Root analytic conductor: |
1.74192 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ380(343,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 380, ( :1/2), −0.564+0.825i)
|
Particular Values
L(1) |
≈ |
0.998164−1.89224i |
L(21) |
≈ |
0.998164−1.89224i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−1.08+0.912i)T |
| 5 | 1+(2.21−0.272i)T |
| 19 | 1−T |
good | 3 | 1+(−1.34+1.34i)T−3iT2 |
| 7 | 1+(0.697+0.697i)T+7iT2 |
| 11 | 1+4.74iT−11T2 |
| 13 | 1+(−3.30−3.30i)T+13iT2 |
| 17 | 1+(1.03−1.03i)T−17iT2 |
| 23 | 1+(−6.40+6.40i)T−23iT2 |
| 29 | 1−1.98iT−29T2 |
| 31 | 1−9.29iT−31T2 |
| 37 | 1+(5.70−5.70i)T−37iT2 |
| 41 | 1−9.41T+41T2 |
| 43 | 1+(1.89−1.89i)T−43iT2 |
| 47 | 1+(1.10+1.10i)T+47iT2 |
| 53 | 1+(0.00450+0.00450i)T+53iT2 |
| 59 | 1−3.02T+59T2 |
| 61 | 1+0.724T+61T2 |
| 67 | 1+(10.1+10.1i)T+67iT2 |
| 71 | 1−9.29iT−71T2 |
| 73 | 1+(4.54+4.54i)T+73iT2 |
| 79 | 1+0.319T+79T2 |
| 83 | 1+(−2.15+2.15i)T−83iT2 |
| 89 | 1−8.13iT−89T2 |
| 97 | 1+(8.29−8.29i)T−97iT2 |
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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
−11.05160045468222067315581543662, −10.65644662635754137018227354485, −8.875381276949514661305089721384, −8.479410069318239158837292322067, −7.05570485938268095938516621156, −6.42519929370706486530778769975, −4.85406277715867640529219910156, −3.60142717864132236624264378653, −2.89796170666020682197711867817, −1.17951039138535198771281343662,
2.85388953043768187694844156477, 3.77821177728813309540648667748, 4.54464187355443901077600344348, 5.72639763710037849009277490969, 7.14825951131535712434626605266, 7.80360250759671102551827886752, 8.849744067609372937814505190644, 9.531922662473790509230248584914, 10.90343948171924302099391672680, 11.83483685590848252285652089764