L(s) = 1 | + (0.255 − 1.39i)2-s + 3.18i·3-s + (−1.86 − 0.711i)4-s + (−1.80 − 1.31i)5-s + (4.43 + 0.815i)6-s + (−1.46 + 2.41i)8-s − 7.15·9-s + (−2.29 + 2.17i)10-s − 4.51i·11-s + (2.26 − 5.95i)12-s + 2.22·13-s + (4.19 − 5.75i)15-s + (2.98 + 2.66i)16-s + 2.52·17-s + (−1.83 + 9.94i)18-s + 5.21·19-s + ⋯ |
L(s) = 1 | + (0.180 − 0.983i)2-s + 1.83i·3-s + (−0.934 − 0.355i)4-s + (−0.808 − 0.588i)5-s + (1.80 + 0.332i)6-s + (−0.519 + 0.854i)8-s − 2.38·9-s + (−0.725 + 0.688i)10-s − 1.36i·11-s + (0.654 − 1.71i)12-s + 0.617·13-s + (1.08 − 1.48i)15-s + (0.746 + 0.665i)16-s + 0.613·17-s + (−0.431 + 2.34i)18-s + 1.19·19-s + ⋯ |
Λ(s)=(=(980s/2ΓC(s)L(s)(0.840+0.541i)Λ(2−s)
Λ(s)=(=(980s/2ΓC(s+1/2)L(s)(0.840+0.541i)Λ(1−s)
Degree: |
2 |
Conductor: |
980
= 22⋅5⋅72
|
Sign: |
0.840+0.541i
|
Analytic conductor: |
7.82533 |
Root analytic conductor: |
2.79738 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ980(979,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 980, ( :1/2), 0.840+0.541i)
|
Particular Values
L(1) |
≈ |
1.19223−0.350714i |
L(21) |
≈ |
1.19223−0.350714i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.255+1.39i)T |
| 5 | 1+(1.80+1.31i)T |
| 7 | 1 |
good | 3 | 1−3.18iT−3T2 |
| 11 | 1+4.51iT−11T2 |
| 13 | 1−2.22T+13T2 |
| 17 | 1−2.52T+17T2 |
| 19 | 1−5.21T+19T2 |
| 23 | 1−1.71T+23T2 |
| 29 | 1+2.31T+29T2 |
| 31 | 1−4.62T+31T2 |
| 37 | 1+0.336iT−37T2 |
| 41 | 1−3.28iT−41T2 |
| 43 | 1−6.66T+43T2 |
| 47 | 1−1.44iT−47T2 |
| 53 | 1+10.0iT−53T2 |
| 59 | 1−3.20T+59T2 |
| 61 | 1−6.05iT−61T2 |
| 67 | 1−11.1T+67T2 |
| 71 | 1+9.15iT−71T2 |
| 73 | 1+3.24T+73T2 |
| 79 | 1+14.2iT−79T2 |
| 83 | 1−11.3iT−83T2 |
| 89 | 1+15.2iT−89T2 |
| 97 | 1−4.49T+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
−10.00457686032921278366635907721, −9.239809335697116153600060978848, −8.661322478104607771712980402867, −7.966940433724020319733207267363, −5.86936814426831182586035266438, −5.24156324256187001776873734003, −4.40182676782255206254831641604, −3.51824904662436765619705519166, −3.12829549831056881529650396869, −0.75494200935979371714091887379,
1.00525572284414680200861721997, 2.65068665978979326012187996768, 3.79371182483405881492737590372, 5.15749914677890920406114887558, 6.14466204535852271295749963415, 6.92703274444332908813517306045, 7.44923547486399711533507110613, 7.898836664625430013278528622627, 8.794773842972641359915223875652, 9.878265446346659954328473483995