L(s) = 1 | + (−0.718 + 1.24i)2-s + (0.967 + 1.67i)4-s + 3.30·5-s + (10.5 − 6.06i)7-s − 8.52·8-s + (−2.37 + 4.11i)10-s + (−7.35 + 12.7i)11-s + (3.48 + 12.5i)13-s + 17.4i·14-s + (2.25 − 3.90i)16-s + (19.9 − 11.4i)17-s + (−9.63 + 5.56i)19-s + (3.19 + 5.54i)20-s + (−10.5 − 18.3i)22-s + (3.03 + 1.75i)23-s + ⋯ |
L(s) = 1 | + (−0.359 + 0.622i)2-s + (0.241 + 0.419i)4-s + 0.661·5-s + (1.50 − 0.866i)7-s − 1.06·8-s + (−0.237 + 0.411i)10-s + (−0.668 + 1.15i)11-s + (0.268 + 0.963i)13-s + 1.24i·14-s + (0.141 − 0.244i)16-s + (1.17 − 0.676i)17-s + (−0.506 + 0.292i)19-s + (0.159 + 0.277i)20-s + (−0.480 − 0.831i)22-s + (0.131 + 0.0761i)23-s + ⋯ |
Λ(s)=(=(117s/2ΓC(s)L(s)(0.324−0.945i)Λ(3−s)
Λ(s)=(=(117s/2ΓC(s+1)L(s)(0.324−0.945i)Λ(1−s)
Degree: |
2 |
Conductor: |
117
= 32⋅13
|
Sign: |
0.324−0.945i
|
Analytic conductor: |
3.18801 |
Root analytic conductor: |
1.78550 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ117(17,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 117, ( :1), 0.324−0.945i)
|
Particular Values
L(23) |
≈ |
1.17713+0.840248i |
L(21) |
≈ |
1.17713+0.840248i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 13 | 1+(−3.48−12.5i)T |
good | 2 | 1+(0.718−1.24i)T+(−2−3.46i)T2 |
| 5 | 1−3.30T+25T2 |
| 7 | 1+(−10.5+6.06i)T+(24.5−42.4i)T2 |
| 11 | 1+(7.35−12.7i)T+(−60.5−104.i)T2 |
| 17 | 1+(−19.9+11.4i)T+(144.5−250.i)T2 |
| 19 | 1+(9.63−5.56i)T+(180.5−312.i)T2 |
| 23 | 1+(−3.03−1.75i)T+(264.5+458.i)T2 |
| 29 | 1+(10.8+6.23i)T+(420.5+728.i)T2 |
| 31 | 1+29.9iT−961T2 |
| 37 | 1+(22.5+13.0i)T+(684.5+1.18e3i)T2 |
| 41 | 1+(−37.9+65.8i)T+(−840.5−1.45e3i)T2 |
| 43 | 1+(−4.34−7.53i)T+(−924.5+1.60e3i)T2 |
| 47 | 1−31.3T+2.20e3T2 |
| 53 | 1−52.3iT−2.80e3T2 |
| 59 | 1+(20.6+35.6i)T+(−1.74e3+3.01e3i)T2 |
| 61 | 1+(−25.1−43.5i)T+(−1.86e3+3.22e3i)T2 |
| 67 | 1+(102.+59.3i)T+(2.24e3+3.88e3i)T2 |
| 71 | 1+(16.8+29.1i)T+(−2.52e3+4.36e3i)T2 |
| 73 | 1−5.21iT−5.32e3T2 |
| 79 | 1+39.7T+6.24e3T2 |
| 83 | 1+141.T+6.88e3T2 |
| 89 | 1+(−11.9+20.7i)T+(−3.96e3−6.85e3i)T2 |
| 97 | 1+(−43.0+24.8i)T+(4.70e3−8.14e3i)T2 |
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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
−13.75145580640151293661189416901, −12.39882946742280462640385673609, −11.42547602190245696011863112783, −10.26429865561453145507132254915, −9.088072462717220562211924655302, −7.73198848019027849280887473022, −7.29031724666143410762043729178, −5.69109280506249585804412492478, −4.25889349466369117048039222329, −2.01410289120608439050958902970,
1.43868524602874116009912958435, 2.85959878445791617715869984083, 5.36687373681717188278096749195, 5.91036910376092960852970102362, 8.035493524072535014172082745962, 8.813709678232729190204876469786, 10.20813169162465585369224981793, 10.90942593561793081400085306334, 11.77454437543789409960041862591, 12.93362456762354824144165408348