L(s) = 1 | + (1.38 − 0.295i)2-s − 1.66i·3-s + (1.82 − 0.817i)4-s + (−2.22 + 0.220i)5-s + (−0.491 − 2.29i)6-s + (2.28 − 1.67i)8-s + 0.236·9-s + (−3.01 + 0.962i)10-s − 4.81i·11-s + (−1.35 − 3.03i)12-s + 2.14·13-s + (0.366 + 3.69i)15-s + (2.66 − 2.98i)16-s − 5.02·17-s + (0.326 − 0.0698i)18-s + 1.36·19-s + ⋯ |
L(s) = 1 | + (0.977 − 0.209i)2-s − 0.959i·3-s + (0.912 − 0.408i)4-s + (−0.995 + 0.0986i)5-s + (−0.200 − 0.938i)6-s + (0.807 − 0.590i)8-s + 0.0787·9-s + (−0.952 + 0.304i)10-s − 1.45i·11-s + (−0.392 − 0.875i)12-s + 0.594·13-s + (0.0946 + 0.955i)15-s + (0.665 − 0.746i)16-s − 1.21·17-s + (0.0770 − 0.0164i)18-s + 0.312·19-s + ⋯ |
Λ(s)=(=(980s/2ΓC(s)L(s)(−0.588+0.808i)Λ(2−s)
Λ(s)=(=(980s/2ΓC(s+1/2)L(s)(−0.588+0.808i)Λ(1−s)
Degree: |
2 |
Conductor: |
980
= 22⋅5⋅72
|
Sign: |
−0.588+0.808i
|
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), −0.588+0.808i)
|
Particular Values
L(1) |
≈ |
1.10016−2.16219i |
L(21) |
≈ |
1.10016−2.16219i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−1.38+0.295i)T |
| 5 | 1+(2.22−0.220i)T |
| 7 | 1 |
good | 3 | 1+1.66iT−3T2 |
| 11 | 1+4.81iT−11T2 |
| 13 | 1−2.14T+13T2 |
| 17 | 1+5.02T+17T2 |
| 19 | 1−1.36T+19T2 |
| 23 | 1+5.18T+23T2 |
| 29 | 1+6.43T+29T2 |
| 31 | 1−4.62T+31T2 |
| 37 | 1−9.82iT−37T2 |
| 41 | 1+4.71iT−41T2 |
| 43 | 1+0.141T+43T2 |
| 47 | 1+2.55iT−47T2 |
| 53 | 1+4.84iT−53T2 |
| 59 | 1−14.1T+59T2 |
| 61 | 1+10.1iT−61T2 |
| 67 | 1−9.64T+67T2 |
| 71 | 1−9.58iT−71T2 |
| 73 | 1−1.67T+73T2 |
| 79 | 1−11.8iT−79T2 |
| 83 | 1+0.811iT−83T2 |
| 89 | 1−16.0iT−89T2 |
| 97 | 1+1.76T+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
−9.942480999667496945428290111568, −8.462651645021202855959487143983, −7.979431529234525053867201693595, −6.87394429436870530795189206374, −6.44434882507246218409984017713, −5.39555956988354582780557726910, −4.15097351485718015513071422028, −3.46535346885381107311563917226, −2.23067182868036723679326509566, −0.825655181589445869522065638230,
2.05024142056754997725837740375, 3.49696458549048418983314751735, 4.30218179851763191247506378564, 4.60464491971378821969996693502, 5.79503385184719709603323742621, 6.98050902497646584582941350981, 7.52054126210171622867196079696, 8.563957804455451967723695319483, 9.565891052752660472356206097250, 10.47017606855606258260373068262