L(s) = 1 | + (0.742 + 0.291i)2-s + (0.844 − 1.51i)3-s + (−0.999 − 0.927i)4-s + (3.49 + 2.38i)5-s + (1.06 − 0.876i)6-s + (−2.63 − 0.242i)7-s + (−1.16 − 2.41i)8-s + (−1.57 − 2.55i)9-s + (1.90 + 2.78i)10-s + (0.481 + 3.19i)11-s + (−2.24 + 0.727i)12-s + (1.47 − 1.17i)13-s + (−1.88 − 0.947i)14-s + (6.55 − 3.27i)15-s + (0.0439 + 0.586i)16-s + (−0.330 − 0.101i)17-s + ⋯ |
L(s) = 1 | + (0.524 + 0.206i)2-s + (0.487 − 0.872i)3-s + (−0.499 − 0.463i)4-s + (1.56 + 1.06i)5-s + (0.435 − 0.357i)6-s + (−0.995 − 0.0917i)7-s + (−0.411 − 0.854i)8-s + (−0.524 − 0.851i)9-s + (0.600 + 0.881i)10-s + (0.145 + 0.963i)11-s + (−0.648 + 0.210i)12-s + (0.410 − 0.327i)13-s + (−0.503 − 0.253i)14-s + (1.69 − 0.844i)15-s + (0.0109 + 0.146i)16-s + (−0.0801 − 0.0247i)17-s + ⋯ |
Λ(s)=(=(147s/2ΓC(s)L(s)(0.914+0.403i)Λ(2−s)
Λ(s)=(=(147s/2ΓC(s+1/2)L(s)(0.914+0.403i)Λ(1−s)
Degree: |
2 |
Conductor: |
147
= 3⋅72
|
Sign: |
0.914+0.403i
|
Analytic conductor: |
1.17380 |
Root analytic conductor: |
1.08342 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ147(101,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 147, ( :1/2), 0.914+0.403i)
|
Particular Values
L(1) |
≈ |
1.56161−0.329149i |
L(21) |
≈ |
1.56161−0.329149i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(−0.844+1.51i)T |
| 7 | 1+(2.63+0.242i)T |
good | 2 | 1+(−0.742−0.291i)T+(1.46+1.36i)T2 |
| 5 | 1+(−3.49−2.38i)T+(1.82+4.65i)T2 |
| 11 | 1+(−0.481−3.19i)T+(−10.5+3.24i)T2 |
| 13 | 1+(−1.47+1.17i)T+(2.89−12.6i)T2 |
| 17 | 1+(0.330+0.101i)T+(14.0+9.57i)T2 |
| 19 | 1+(3.79−2.19i)T+(9.5−16.4i)T2 |
| 23 | 1+(−0.617−2.00i)T+(−19.0+12.9i)T2 |
| 29 | 1+(3.02−0.690i)T+(26.1−12.5i)T2 |
| 31 | 1+(6.61+3.81i)T+(15.5+26.8i)T2 |
| 37 | 1+(−4.19+3.88i)T+(2.76−36.8i)T2 |
| 41 | 1+(−5.35+2.58i)T+(25.5−32.0i)T2 |
| 43 | 1+(1.13+0.545i)T+(26.8+33.6i)T2 |
| 47 | 1+(0.205−0.523i)T+(−34.4−31.9i)T2 |
| 53 | 1+(0.0176−0.0190i)T+(−3.96−52.8i)T2 |
| 59 | 1+(−2.74+1.87i)T+(21.5−54.9i)T2 |
| 61 | 1+(4.20+4.53i)T+(−4.55+60.8i)T2 |
| 67 | 1+(−6.45+11.1i)T+(−33.5−58.0i)T2 |
| 71 | 1+(−7.01−1.60i)T+(63.9+30.8i)T2 |
| 73 | 1+(2.47−0.972i)T+(53.5−49.6i)T2 |
| 79 | 1+(1.49+2.58i)T+(−39.5+68.4i)T2 |
| 83 | 1+(−5.31+6.66i)T+(−18.4−80.9i)T2 |
| 89 | 1+(1.88+0.284i)T+(85.0+26.2i)T2 |
| 97 | 1+11.3iT−97T2 |
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
−13.03374277477196109200151249889, −12.75405713080069043083692174264, −10.76757535847554785946841871617, −9.707561947587011168033473133296, −9.217492453926486480611222021817, −7.24644725104330220106645028879, −6.34620297298104769727044824117, −5.74260319161519100005625326063, −3.56920722644674155297767972073, −2.06533037920862605213729691606,
2.62558687860289765025031630144, 4.01487436891207547417813785423, 5.21861274803070097193901292554, 6.14341680075367651153248551982, 8.545166922485943664577632889437, 9.012211126927867068647327062224, 9.773224546239354613394821583299, 11.02828593510534961799654757757, 12.59698082161247324244190540751, 13.28494360636994532382527317988