L(s) = 1 | + (0.391 + 1.20i)2-s + (0.458 − 1.41i)3-s + (0.320 − 0.232i)4-s − 3.80·5-s + 1.88·6-s + (−1.77 + 1.28i)7-s + (2.45 + 1.78i)8-s + (0.644 + 0.468i)9-s + (−1.48 − 4.57i)10-s + (0.769 − 0.558i)11-s + (−0.181 − 0.558i)12-s + (0.0519 − 0.159i)13-s + (−2.24 − 1.62i)14-s + (−1.74 + 5.36i)15-s + (−0.943 + 2.90i)16-s + (5.32 + 3.86i)17-s + ⋯ |
L(s) = 1 | + (0.276 + 0.851i)2-s + (0.264 − 0.815i)3-s + (0.160 − 0.116i)4-s − 1.69·5-s + 0.767·6-s + (−0.669 + 0.486i)7-s + (0.867 + 0.630i)8-s + (0.214 + 0.156i)9-s + (−0.470 − 1.44i)10-s + (0.231 − 0.168i)11-s + (−0.0523 − 0.161i)12-s + (0.0144 − 0.0443i)13-s + (−0.599 − 0.435i)14-s + (−0.450 + 1.38i)15-s + (−0.235 + 0.725i)16-s + (1.29 + 0.937i)17-s + ⋯ |
Λ(s)=(=(961s/2ΓC(s)L(s)(0.501−0.865i)Λ(2−s)
Λ(s)=(=(961s/2ΓC(s+1/2)L(s)(0.501−0.865i)Λ(1−s)
Degree: |
2 |
Conductor: |
961
= 312
|
Sign: |
0.501−0.865i
|
Analytic conductor: |
7.67362 |
Root analytic conductor: |
2.77013 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ961(531,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 961, ( :1/2), 0.501−0.865i)
|
Particular Values
L(1) |
≈ |
1.46860+0.846423i |
L(21) |
≈ |
1.46860+0.846423i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 31 | 1 |
good | 2 | 1+(−0.391−1.20i)T+(−1.61+1.17i)T2 |
| 3 | 1+(−0.458+1.41i)T+(−2.42−1.76i)T2 |
| 5 | 1+3.80T+5T2 |
| 7 | 1+(1.77−1.28i)T+(2.16−6.65i)T2 |
| 11 | 1+(−0.769+0.558i)T+(3.39−10.4i)T2 |
| 13 | 1+(−0.0519+0.159i)T+(−10.5−7.64i)T2 |
| 17 | 1+(−5.32−3.86i)T+(5.25+16.1i)T2 |
| 19 | 1+(0.356+1.09i)T+(−15.3+11.1i)T2 |
| 23 | 1+(−3.74−2.72i)T+(7.10+21.8i)T2 |
| 29 | 1+(−0.413−1.27i)T+(−23.4+17.0i)T2 |
| 37 | 1+3.87T+37T2 |
| 41 | 1+(−0.101−0.312i)T+(−33.1+24.0i)T2 |
| 43 | 1+(−2.97−9.15i)T+(−34.7+25.2i)T2 |
| 47 | 1+(−1.74+5.36i)T+(−38.0−27.6i)T2 |
| 53 | 1+(−5.93−4.31i)T+(16.3+50.4i)T2 |
| 59 | 1+(0.819−2.52i)T+(−47.7−34.6i)T2 |
| 61 | 1+1.74T+61T2 |
| 67 | 1−0.552T+67T2 |
| 71 | 1+(0.919+0.668i)T+(21.9+67.5i)T2 |
| 73 | 1+(−6.40+4.65i)T+(22.5−69.4i)T2 |
| 79 | 1+(3.67+2.66i)T+(24.4+75.1i)T2 |
| 83 | 1+(−0.100−0.310i)T+(−67.1+48.7i)T2 |
| 89 | 1+(−11.9+8.64i)T+(27.5−84.6i)T2 |
| 97 | 1+(12.5−9.12i)T+(29.9−92.2i)T2 |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−10.28283359697501905323184405398, −8.929136133135624405578863712825, −8.043274896546945712986944866052, −7.59795075587895492911180740981, −6.92113837144512785180013713152, −6.12511618370272725087871101062, −5.05653892291422283628166090283, −3.93950194209180992917566330743, −2.89582085733345213315121162548, −1.27749348790612967915597221255,
0.835431649903740984266659843376, 2.89688837546857195956555442661, 3.63802890890178464906632472779, 4.04322115518178739833087516718, 4.99481261369495254096339867018, 6.86845495317616279333615345225, 7.27753758383668985290126306329, 8.224647989473540982031524574620, 9.305778002964025298047167691104, 10.11720112452119281228754758209