L(s) = 1 | + (0.596 + 0.0240i)2-s + (0.703 − 2.43i)3-s + (−1.63 − 0.132i)4-s + (3.70 + 1.40i)5-s + (0.478 − 1.43i)6-s + (−0.324 − 0.762i)7-s + (−2.16 − 0.262i)8-s + (−2.87 − 1.81i)9-s + (2.17 + 0.926i)10-s + (−0.896 − 1.41i)11-s + (−1.47 + 3.88i)12-s + (−2.52 + 2.57i)13-s + (−0.175 − 0.462i)14-s + (6.01 − 8.00i)15-s + (1.96 + 0.318i)16-s + (6.43 − 2.74i)17-s + ⋯ |
L(s) = 1 | + (0.422 + 0.0170i)2-s + (0.406 − 1.40i)3-s + (−0.818 − 0.0661i)4-s + (1.65 + 0.627i)5-s + (0.195 − 0.585i)6-s + (−0.122 − 0.288i)7-s + (−0.763 − 0.0927i)8-s + (−0.958 − 0.605i)9-s + (0.687 + 0.292i)10-s + (−0.270 − 0.427i)11-s + (−0.425 + 1.12i)12-s + (−0.699 + 0.714i)13-s + (−0.0469 − 0.123i)14-s + (1.55 − 2.06i)15-s + (0.490 + 0.0796i)16-s + (1.56 − 0.665i)17-s + ⋯ |
Λ(s)=(=(169s/2ΓC(s)L(s)(0.530+0.847i)Λ(2−s)
Λ(s)=(=(169s/2ΓC(s+1/2)L(s)(0.530+0.847i)Λ(1−s)
Degree: |
2 |
Conductor: |
169
= 132
|
Sign: |
0.530+0.847i
|
Analytic conductor: |
1.34947 |
Root analytic conductor: |
1.16166 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ169(4,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 169, ( :1/2), 0.530+0.847i)
|
Particular Values
L(1) |
≈ |
1.35928−0.752856i |
L(21) |
≈ |
1.35928−0.752856i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 13 | 1+(2.52−2.57i)T |
good | 2 | 1+(−0.596−0.0240i)T+(1.99+0.160i)T2 |
| 3 | 1+(−0.703+2.43i)T+(−2.53−1.60i)T2 |
| 5 | 1+(−3.70−1.40i)T+(3.74+3.31i)T2 |
| 7 | 1+(0.324+0.762i)T+(−4.84+5.04i)T2 |
| 11 | 1+(0.896+1.41i)T+(−4.71+9.93i)T2 |
| 17 | 1+(−6.43+2.74i)T+(11.7−12.2i)T2 |
| 19 | 1+(6.17−3.56i)T+(9.5−16.4i)T2 |
| 23 | 1+(1.23−2.13i)T+(−11.5−19.9i)T2 |
| 29 | 1+(0.333−8.28i)T+(−28.9−2.33i)T2 |
| 31 | 1+(−0.422−0.476i)T+(−3.73+30.7i)T2 |
| 37 | 1+(4.24−0.865i)T+(34.0−14.5i)T2 |
| 41 | 1+(3.37+0.977i)T+(34.6+21.9i)T2 |
| 43 | 1+(−0.218+1.07i)T+(−39.5−16.8i)T2 |
| 47 | 1+(1.62−1.11i)T+(16.6−43.9i)T2 |
| 53 | 1+(0.146−1.21i)T+(−51.4−12.6i)T2 |
| 59 | 1+(1.82+11.2i)T+(−55.9+18.6i)T2 |
| 61 | 1+(1.57−1.18i)T+(16.9−58.5i)T2 |
| 67 | 1+(0.248+3.07i)T+(−66.1+10.7i)T2 |
| 71 | 1+(2.34−2.24i)T+(2.85−70.9i)T2 |
| 73 | 1+(−4.05+7.71i)T+(−41.4−60.0i)T2 |
| 79 | 1+(3.06+4.44i)T+(−28.0+73.8i)T2 |
| 83 | 1+(1.84−7.49i)T+(−73.4−38.5i)T2 |
| 89 | 1+(14.6+8.44i)T+(44.5+77.0i)T2 |
| 97 | 1+(0.0230+0.0188i)T+(19.4+95.0i)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
−12.85763581531037432616539492165, −12.19707425058941366029858197294, −10.41509626862670720307827880588, −9.603431327959235573532882140094, −8.498968384885938443360378988380, −7.18781998446995535484314650179, −6.24357055998728091756393597880, −5.28452575154974425133305177007, −3.16778407750757940894655670948, −1.74628867828756993236492666591,
2.67282562755406300532057182839, 4.26945292368217674609900172383, 5.17526434169166277470411324840, 5.93080903854298529010410017269, 8.320537498078993781348396825121, 9.144921965515437066481701437403, 9.998588179703306840586682935554, 10.26657428696973952293999719403, 12.38844250300988720927870545413, 13.01014485169274814415911187934