L(s) = 1 | + (0.258 − 0.965i)2-s + (1.71 + 0.208i)3-s + (−0.866 − 0.499i)4-s + (0.406 − 1.51i)5-s + (0.646 − 1.60i)6-s + (−3.31 − 3.31i)7-s + (−0.707 + 0.707i)8-s + (2.91 + 0.718i)9-s + (−1.36 − 0.785i)10-s + (−1.47 − 0.395i)11-s + (−1.38 − 1.04i)12-s + (3.50 + 0.853i)13-s + (−4.05 + 2.34i)14-s + (1.01 − 2.52i)15-s + (0.500 + 0.866i)16-s + (2.99 + 5.18i)17-s + ⋯ |
L(s) = 1 | + (0.183 − 0.683i)2-s + (0.992 + 0.120i)3-s + (−0.433 − 0.249i)4-s + (0.181 − 0.678i)5-s + (0.264 − 0.655i)6-s + (−1.25 − 1.25i)7-s + (−0.249 + 0.249i)8-s + (0.970 + 0.239i)9-s + (−0.430 − 0.248i)10-s + (−0.445 − 0.119i)11-s + (−0.399 − 0.300i)12-s + (0.971 + 0.236i)13-s + (−1.08 + 0.626i)14-s + (0.262 − 0.651i)15-s + (0.125 + 0.216i)16-s + (0.725 + 1.25i)17-s + ⋯ |
Λ(s)=(=(234s/2ΓC(s)L(s)(0.0914+0.995i)Λ(2−s)
Λ(s)=(=(234s/2ΓC(s+1/2)L(s)(0.0914+0.995i)Λ(1−s)
Degree: |
2 |
Conductor: |
234
= 2⋅32⋅13
|
Sign: |
0.0914+0.995i
|
Analytic conductor: |
1.86849 |
Root analytic conductor: |
1.36693 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ234(227,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 234, ( :1/2), 0.0914+0.995i)
|
Particular Values
L(1) |
≈ |
1.21566−1.10911i |
L(21) |
≈ |
1.21566−1.10911i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.258+0.965i)T |
| 3 | 1+(−1.71−0.208i)T |
| 13 | 1+(−3.50−0.853i)T |
good | 5 | 1+(−0.406+1.51i)T+(−4.33−2.5i)T2 |
| 7 | 1+(3.31+3.31i)T+7iT2 |
| 11 | 1+(1.47+0.395i)T+(9.52+5.5i)T2 |
| 17 | 1+(−2.99−5.18i)T+(−8.5+14.7i)T2 |
| 19 | 1+(−5.70−1.52i)T+(16.4+9.5i)T2 |
| 23 | 1+4.97T+23T2 |
| 29 | 1+(5.94−3.43i)T+(14.5−25.1i)T2 |
| 31 | 1+(−1.01−0.271i)T+(26.8+15.5i)T2 |
| 37 | 1+(6.94−1.86i)T+(32.0−18.5i)T2 |
| 41 | 1+(3.66+3.66i)T+41iT2 |
| 43 | 1+12.2iT−43T2 |
| 47 | 1+(−1.64−6.14i)T+(−40.7+23.5i)T2 |
| 53 | 1+2.78iT−53T2 |
| 59 | 1+(−1.54−5.76i)T+(−51.0+29.5i)T2 |
| 61 | 1−4.37T+61T2 |
| 67 | 1+(−3.96+3.96i)T−67iT2 |
| 71 | 1+(0.0745−0.278i)T+(−61.4−35.5i)T2 |
| 73 | 1+(−0.964−0.964i)T+73iT2 |
| 79 | 1+(0.0673−0.116i)T+(−39.5−68.4i)T2 |
| 83 | 1+(11.2−3.00i)T+(71.8−41.5i)T2 |
| 89 | 1+(1.46+5.45i)T+(−77.0+44.5i)T2 |
| 97 | 1+(4.71−4.71i)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
−12.27936144535218744328192941694, −10.62806658488448450981329696087, −10.09156833217167157529909038259, −9.204800569986487588175611559687, −8.227905205967869010422969489783, −7.06186618148774000583044551447, −5.56314352078114110706706148966, −3.88858975795369111121624779843, −3.42287270145729047395033016287, −1.41889348162327999401376958270,
2.70909476653849312932305038513, 3.46056925315022265597811986672, 5.39948396619920582191676626873, 6.41485072025295394748270518274, 7.37800616515877409192310973188, 8.425484161705222946086331772544, 9.462106898541380580068101404513, 9.957878463539164623777835711625, 11.68509967343838973134487855674, 12.70668631654421638202655594730