L(s) = 1 | + (0.928 + 1.60i)3-s + (−1.51 + 1.51i)5-s + (−2.97 + 0.797i)7-s + (−0.223 + 0.386i)9-s + (−0.865 − 0.231i)11-s + (0.159 + 3.60i)13-s + (−3.84 − 1.03i)15-s + (1.05 + 0.610i)17-s + (−6.68 + 1.79i)19-s + (−4.04 − 4.04i)21-s + (0.433 + 0.751i)23-s + 0.390i·25-s + 4.74·27-s + (−3.26 + 1.88i)29-s + (5.06 − 5.06i)31-s + ⋯ |
L(s) = 1 | + (0.535 + 0.928i)3-s + (−0.678 + 0.678i)5-s + (−1.12 + 0.301i)7-s + (−0.0743 + 0.128i)9-s + (−0.260 − 0.0699i)11-s + (0.0442 + 0.999i)13-s + (−0.994 − 0.266i)15-s + (0.256 + 0.147i)17-s + (−1.53 + 0.410i)19-s + (−0.883 − 0.883i)21-s + (0.0904 + 0.156i)23-s + 0.0780i·25-s + 0.912·27-s + (−0.606 + 0.350i)29-s + (0.909 − 0.909i)31-s + ⋯ |
Λ(s)=(=(416s/2ΓC(s)L(s)(−0.836−0.547i)Λ(2−s)
Λ(s)=(=(416s/2ΓC(s+1/2)L(s)(−0.836−0.547i)Λ(1−s)
Degree: |
2 |
Conductor: |
416
= 25⋅13
|
Sign: |
−0.836−0.547i
|
Analytic conductor: |
3.32177 |
Root analytic conductor: |
1.82257 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ416(271,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 416, ( :1/2), −0.836−0.547i)
|
Particular Values
L(1) |
≈ |
0.282816+0.949214i |
L(21) |
≈ |
0.282816+0.949214i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 13 | 1+(−0.159−3.60i)T |
good | 3 | 1+(−0.928−1.60i)T+(−1.5+2.59i)T2 |
| 5 | 1+(1.51−1.51i)T−5iT2 |
| 7 | 1+(2.97−0.797i)T+(6.06−3.5i)T2 |
| 11 | 1+(0.865+0.231i)T+(9.52+5.5i)T2 |
| 17 | 1+(−1.05−0.610i)T+(8.5+14.7i)T2 |
| 19 | 1+(6.68−1.79i)T+(16.4−9.5i)T2 |
| 23 | 1+(−0.433−0.751i)T+(−11.5+19.9i)T2 |
| 29 | 1+(3.26−1.88i)T+(14.5−25.1i)T2 |
| 31 | 1+(−5.06+5.06i)T−31iT2 |
| 37 | 1+(−2.52+9.43i)T+(−32.0−18.5i)T2 |
| 41 | 1+(3.12−11.6i)T+(−35.5−20.5i)T2 |
| 43 | 1+(−4.22−2.44i)T+(21.5+37.2i)T2 |
| 47 | 1+(−4.24−4.24i)T+47iT2 |
| 53 | 1−2.16iT−53T2 |
| 59 | 1+(0.0382+0.142i)T+(−51.0+29.5i)T2 |
| 61 | 1+(−7.26−4.19i)T+(30.5+52.8i)T2 |
| 67 | 1+(0.422−1.57i)T+(−58.0−33.5i)T2 |
| 71 | 1+(−4.27−15.9i)T+(−61.4+35.5i)T2 |
| 73 | 1+(−9.55+9.55i)T−73iT2 |
| 79 | 1+6.37iT−79T2 |
| 83 | 1+(3.53+3.53i)T+83iT2 |
| 89 | 1+(−9.33−2.50i)T+(77.0+44.5i)T2 |
| 97 | 1+(10.6−2.86i)T+(84.0−48.5i)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
−11.38764246548789437148579454738, −10.58533010989516777108197337417, −9.679217058099894734141582734341, −9.094294992364563831452115053389, −8.020398560411429931563490701016, −6.85815375726010592225065996246, −6.03217806675435271001289001740, −4.33527791755000266055799893590, −3.66432876481408207783649804122, −2.62037520448322280606885243100,
0.58425726421003469668418697927, 2.44476016849574478644300608231, 3.65919645881682509264509302341, 4.93613017296035212807327182854, 6.36772196808951909708347015785, 7.18276507894847054547160498531, 8.161556968089672830096835902865, 8.661068699279373549014132692137, 9.971210224866643484570630443669, 10.74682628725668672945377490983