L(s) = 1 | + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (1.11 − 3.44i)5-s + (3.11 − 2.26i)7-s + (−0.809 − 0.587i)8-s + 3.61·10-s + (−3.23 − 0.726i)11-s + (2 + 6.15i)13-s + (3.11 + 2.26i)14-s + (0.309 − 0.951i)16-s + (−1 + 3.07i)17-s + (−1.61 − 1.17i)19-s + (1.11 + 3.44i)20-s + (−0.309 − 3.30i)22-s + 2.76·23-s + ⋯ |
L(s) = 1 | + (0.218 + 0.672i)2-s + (−0.404 + 0.293i)4-s + (0.499 − 1.53i)5-s + (1.17 − 0.856i)7-s + (−0.286 − 0.207i)8-s + 1.14·10-s + (−0.975 − 0.219i)11-s + (0.554 + 1.70i)13-s + (0.833 + 0.605i)14-s + (0.0772 − 0.237i)16-s + (−0.242 + 0.746i)17-s + (−0.371 − 0.269i)19-s + (0.249 + 0.769i)20-s + (−0.0658 − 0.704i)22-s + 0.576·23-s + ⋯ |
Λ(s)=(=(198s/2ΓC(s)L(s)(0.998−0.0475i)Λ(2−s)
Λ(s)=(=(198s/2ΓC(s+1/2)L(s)(0.998−0.0475i)Λ(1−s)
Degree: |
2 |
Conductor: |
198
= 2⋅32⋅11
|
Sign: |
0.998−0.0475i
|
Analytic conductor: |
1.58103 |
Root analytic conductor: |
1.25739 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ198(181,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 198, ( :1/2), 0.998−0.0475i)
|
Particular Values
L(1) |
≈ |
1.43092+0.0340250i |
L(21) |
≈ |
1.43092+0.0340250i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.309−0.951i)T |
| 3 | 1 |
| 11 | 1+(3.23+0.726i)T |
good | 5 | 1+(−1.11+3.44i)T+(−4.04−2.93i)T2 |
| 7 | 1+(−3.11+2.26i)T+(2.16−6.65i)T2 |
| 13 | 1+(−2−6.15i)T+(−10.5+7.64i)T2 |
| 17 | 1+(1−3.07i)T+(−13.7−9.99i)T2 |
| 19 | 1+(1.61+1.17i)T+(5.87+18.0i)T2 |
| 23 | 1−2.76T+23T2 |
| 29 | 1+(0.309−0.224i)T+(8.96−27.5i)T2 |
| 31 | 1+(−0.0450−0.138i)T+(−25.0+18.2i)T2 |
| 37 | 1+(3−2.17i)T+(11.4−35.1i)T2 |
| 41 | 1+(−5.47−3.97i)T+(12.6+38.9i)T2 |
| 43 | 1+43T2 |
| 47 | 1+(3.85+2.80i)T+(14.5+44.6i)T2 |
| 53 | 1+(−2.89−8.92i)T+(−42.8+31.1i)T2 |
| 59 | 1+(9.97−7.24i)T+(18.2−56.1i)T2 |
| 61 | 1+(−0.854+2.62i)T+(−49.3−35.8i)T2 |
| 67 | 1+10.9T+67T2 |
| 71 | 1+(0.236−0.726i)T+(−57.4−41.7i)T2 |
| 73 | 1+(−1.92+1.40i)T+(22.5−69.4i)T2 |
| 79 | 1+(2.20+6.79i)T+(−63.9+46.4i)T2 |
| 83 | 1+(−2.40+7.41i)T+(−67.1−48.7i)T2 |
| 89 | 1+6.18T+89T2 |
| 97 | 1+(2.20+6.79i)T+(−78.4+57.0i)T2 |
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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
−12.81711256754345927920658498578, −11.60601211684589455213068036080, −10.54853828869623799796987945448, −9.097836286281403845699204782829, −8.505737784214765755569863843525, −7.50145453927126694687091429115, −6.09986302353272208305694498552, −4.85594093046387365551415191371, −4.31800875702691473113706808648, −1.53576062301919089794728346888,
2.24743704848796746777231755689, 3.13821936892743755580842546584, 5.08747145538125390610482919191, 5.92644019050461093026748684200, 7.46104389249162055933575148093, 8.475433399714909098117673040641, 9.888400903693581023387189339465, 10.80692829784314327378595534410, 11.14756528696380628743561584471, 12.43910069691581540292117441542