L(s) = 1 | + (0.640 + 0.465i)2-s + (0.309 − 0.951i)3-s + (−0.424 − 1.30i)4-s + (−2.72 + 1.98i)5-s + (0.640 − 0.465i)6-s + (−0.780 − 2.40i)7-s + (0.825 − 2.54i)8-s + (−0.809 − 0.587i)9-s − 2.67·10-s − 1.37·12-s + (−4.72 − 3.43i)13-s + (0.618 − 1.90i)14-s + (1.04 + 3.20i)15-s + (−0.507 + 0.368i)16-s + (2.16 − 1.57i)17-s + (−0.244 − 0.753i)18-s + ⋯ |
L(s) = 1 | + (0.453 + 0.329i)2-s + (0.178 − 0.549i)3-s + (−0.212 − 0.652i)4-s + (−1.22 + 0.886i)5-s + (0.261 − 0.190i)6-s + (−0.294 − 0.907i)7-s + (0.291 − 0.898i)8-s + (−0.269 − 0.195i)9-s − 0.844·10-s − 0.396·12-s + (−1.31 − 0.952i)13-s + (0.165 − 0.508i)14-s + (0.269 + 0.828i)15-s + (−0.126 + 0.0922i)16-s + (0.524 − 0.380i)17-s + (−0.0577 − 0.177i)18-s + ⋯ |
Λ(s)=(=(363s/2ΓC(s)L(s)(−0.469+0.882i)Λ(2−s)
Λ(s)=(=(363s/2ΓC(s+1/2)L(s)(−0.469+0.882i)Λ(1−s)
Degree: |
2 |
Conductor: |
363
= 3⋅112
|
Sign: |
−0.469+0.882i
|
Analytic conductor: |
2.89856 |
Root analytic conductor: |
1.70251 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ363(202,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 363, ( :1/2), −0.469+0.882i)
|
Particular Values
L(1) |
≈ |
0.487257−0.811327i |
L(21) |
≈ |
0.487257−0.811327i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(−0.309+0.951i)T |
| 11 | 1 |
good | 2 | 1+(−0.640−0.465i)T+(0.618+1.90i)T2 |
| 5 | 1+(2.72−1.98i)T+(1.54−4.75i)T2 |
| 7 | 1+(0.780+2.40i)T+(−5.66+4.11i)T2 |
| 13 | 1+(4.72+3.43i)T+(4.01+12.3i)T2 |
| 17 | 1+(−2.16+1.57i)T+(5.25−16.1i)T2 |
| 19 | 1+(−0.290+0.893i)T+(−15.3−11.1i)T2 |
| 23 | 1−2T+23T2 |
| 29 | 1+(−0.244−0.753i)T+(−23.4+17.0i)T2 |
| 31 | 1+(1.31+0.956i)T+(9.57+29.4i)T2 |
| 37 | 1+(−1.54−4.75i)T+(−29.9+21.7i)T2 |
| 41 | 1+(−3.36+10.3i)T+(−33.1−24.0i)T2 |
| 43 | 1−6.63T+43T2 |
| 47 | 1+(3.93−12.1i)T+(−38.0−27.6i)T2 |
| 53 | 1+(−3.33−2.41i)T+(16.3+50.4i)T2 |
| 59 | 1+(1.85+5.70i)T+(−47.7+34.6i)T2 |
| 61 | 1+(−4.84+3.51i)T+(18.8−58.0i)T2 |
| 67 | 1+1.11T+67T2 |
| 71 | 1+(−8.69+6.31i)T+(21.9−67.5i)T2 |
| 73 | 1+(2.82+8.70i)T+(−59.0+42.9i)T2 |
| 79 | 1+(−3.32−2.41i)T+(24.4+75.1i)T2 |
| 83 | 1+(1.52−1.10i)T+(25.6−78.9i)T2 |
| 89 | 1+0.627T+89T2 |
| 97 | 1+(8.48+6.16i)T+(29.9+92.2i)T2 |
show more | |
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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
−11.03572820982431825611533999411, −10.36844518623831580701042809717, −9.408715212549200981601070532859, −7.75294038368010207311574088057, −7.36017471726892485910620323785, −6.53300201701359643232738048421, −5.20228016391967731989204452068, −4.02232239845444830468545885196, −2.93766257978848011739561374825, −0.54342189078466628748981980560,
2.53898598041433881755967970756, 3.78545087316832722549747445997, 4.54837480578173902220939230639, 5.42890037863529505318734439049, 7.25923895911084520143787519428, 8.185337135395066024267705511938, 8.887116730131215245537041386624, 9.700030715122904142628911612950, 11.21211595636438214027286984362, 11.97663768300446330777269044560