L(s) = 1 | + (−1.23 + 0.262i)3-s + (−0.367 − 3.50i)5-s + (0.745 + 2.53i)7-s + (−1.28 + 0.573i)9-s + (0.496 − 3.27i)11-s + (−5.68 − 4.13i)13-s + (1.37 + 4.22i)15-s + (0.124 + 0.0554i)17-s + (−3.02 − 3.36i)19-s + (−1.58 − 2.93i)21-s + (0.770 − 1.33i)23-s + (−7.22 + 1.53i)25-s + (4.49 − 3.26i)27-s + (−0.662 − 2.04i)29-s + (−0.0801 + 0.762i)31-s + ⋯ |
L(s) = 1 | + (−0.712 + 0.151i)3-s + (−0.164 − 1.56i)5-s + (0.281 + 0.959i)7-s + (−0.429 + 0.191i)9-s + (0.149 − 0.988i)11-s + (−1.57 − 1.14i)13-s + (0.354 + 1.08i)15-s + (0.0301 + 0.0134i)17-s + (−0.694 − 0.770i)19-s + (−0.345 − 0.640i)21-s + (0.160 − 0.278i)23-s + (−1.44 + 0.307i)25-s + (0.865 − 0.629i)27-s + (−0.123 − 0.378i)29-s + (−0.0144 + 0.137i)31-s + ⋯ |
Λ(s)=(=(308s/2ΓC(s)L(s)(−0.577+0.816i)Λ(2−s)
Λ(s)=(=(308s/2ΓC(s+1/2)L(s)(−0.577+0.816i)Λ(1−s)
Degree: |
2 |
Conductor: |
308
= 22⋅7⋅11
|
Sign: |
−0.577+0.816i
|
Analytic conductor: |
2.45939 |
Root analytic conductor: |
1.56824 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ308(37,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 308, ( :1/2), −0.577+0.816i)
|
Particular Values
L(1) |
≈ |
0.282080−0.544906i |
L(21) |
≈ |
0.282080−0.544906i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 7 | 1+(−0.745−2.53i)T |
| 11 | 1+(−0.496+3.27i)T |
good | 3 | 1+(1.23−0.262i)T+(2.74−1.22i)T2 |
| 5 | 1+(0.367+3.50i)T+(−4.89+1.03i)T2 |
| 13 | 1+(5.68+4.13i)T+(4.01+12.3i)T2 |
| 17 | 1+(−0.124−0.0554i)T+(11.3+12.6i)T2 |
| 19 | 1+(3.02+3.36i)T+(−1.98+18.8i)T2 |
| 23 | 1+(−0.770+1.33i)T+(−11.5−19.9i)T2 |
| 29 | 1+(0.662+2.04i)T+(−23.4+17.0i)T2 |
| 31 | 1+(0.0801−0.762i)T+(−30.3−6.44i)T2 |
| 37 | 1+(−2.61−0.556i)T+(33.8+15.0i)T2 |
| 41 | 1+(−2.06+6.36i)T+(−33.1−24.0i)T2 |
| 43 | 1+2.40T+43T2 |
| 47 | 1+(−4.98−5.54i)T+(−4.91+46.7i)T2 |
| 53 | 1+(1.12−10.7i)T+(−51.8−11.0i)T2 |
| 59 | 1+(2.24−2.48i)T+(−6.16−58.6i)T2 |
| 61 | 1+(0.0132+0.125i)T+(−59.6+12.6i)T2 |
| 67 | 1+(−4.39−7.61i)T+(−33.5+58.0i)T2 |
| 71 | 1+(−4.94+3.59i)T+(21.9−67.5i)T2 |
| 73 | 1+(−10.0+11.1i)T+(−7.63−72.6i)T2 |
| 79 | 1+(−6.92+3.08i)T+(52.8−58.7i)T2 |
| 83 | 1+(6.56−4.76i)T+(25.6−78.9i)T2 |
| 89 | 1+(−7.57+13.1i)T+(−44.5−77.0i)T2 |
| 97 | 1+(9.61+6.98i)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
−11.53428991519048173243936959866, −10.60105977133254498416221605150, −9.287415881306250287689659226821, −8.607371091957525928803352792197, −7.79637844058091195613813914619, −6.00710949999057903547337632923, −5.28914568089900236732856605546, −4.63139125140298897768925973037, −2.62598603122645913666021579858, −0.46519864889948653582168416803,
2.22964498469768770228607632669, 3.78099600337908795959031341655, 4.96654125518292848088305143610, 6.54730575533727760860234506770, 6.94201987893355145373352430998, 7.83167958080478694702141298256, 9.576400308188744120173814826550, 10.29457222665088312153813772559, 11.16964532293724442814900433524, 11.77208959054421279812129686130