L(s) = 1 | + (−0.834 + 3.11i)2-s + (1.02 + 1.77i)3-s + (−5.54 − 3.19i)4-s + (−5.50 − 5.50i)5-s + (−6.37 + 1.70i)6-s + (−0.846 − 3.16i)7-s + (5.47 − 5.47i)8-s + (2.40 − 4.16i)9-s + (21.7 − 12.5i)10-s + (−4.90 − 1.31i)11-s − 13.0i·12-s + 10.5·14-s + (4.12 − 15.3i)15-s + (−0.322 − 0.558i)16-s + (−8.72 − 5.03i)17-s + (10.9 + 10.9i)18-s + ⋯ |
L(s) = 1 | + (−0.417 + 1.55i)2-s + (0.341 + 0.590i)3-s + (−1.38 − 0.799i)4-s + (−1.10 − 1.10i)5-s + (−1.06 + 0.284i)6-s + (−0.120 − 0.451i)7-s + (0.683 − 0.683i)8-s + (0.267 − 0.463i)9-s + (2.17 − 1.25i)10-s + (−0.445 − 0.119i)11-s − 1.09i·12-s + 0.753·14-s + (0.274 − 1.02i)15-s + (−0.0201 − 0.0349i)16-s + (−0.513 − 0.296i)17-s + (0.609 + 0.609i)18-s + ⋯ |
Λ(s)=(=(169s/2ΓC(s)L(s)(0.750+0.661i)Λ(3−s)
Λ(s)=(=(169s/2ΓC(s+1)L(s)(0.750+0.661i)Λ(1−s)
Degree: |
2 |
Conductor: |
169
= 132
|
Sign: |
0.750+0.661i
|
Analytic conductor: |
4.60491 |
Root analytic conductor: |
2.14590 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ169(89,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 169, ( :1), 0.750+0.661i)
|
Particular Values
L(23) |
≈ |
0.346603−0.130927i |
L(21) |
≈ |
0.346603−0.130927i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 13 | 1 |
good | 2 | 1+(0.834−3.11i)T+(−3.46−2i)T2 |
| 3 | 1+(−1.02−1.77i)T+(−4.5+7.79i)T2 |
| 5 | 1+(5.50+5.50i)T+25iT2 |
| 7 | 1+(0.846+3.16i)T+(−42.4+24.5i)T2 |
| 11 | 1+(4.90+1.31i)T+(104.+60.5i)T2 |
| 17 | 1+(8.72+5.03i)T+(144.5+250.i)T2 |
| 19 | 1+(11.4−3.07i)T+(312.−180.5i)T2 |
| 23 | 1+(27.5−15.9i)T+(264.5−458.i)T2 |
| 29 | 1+(16.3+28.2i)T+(−420.5+728.i)T2 |
| 31 | 1+(8.05+8.05i)T+961iT2 |
| 37 | 1+(−42.7−11.4i)T+(1.18e3+684.5i)T2 |
| 41 | 1+(−12.2+45.6i)T+(−1.45e3−840.5i)T2 |
| 43 | 1+(20.1+11.6i)T+(924.5+1.60e3i)T2 |
| 47 | 1+(64.6−64.6i)T−2.20e3iT2 |
| 53 | 1+48.7T+2.80e3T2 |
| 59 | 1+(6.31+23.5i)T+(−3.01e3+1.74e3i)T2 |
| 61 | 1+(6.30−10.9i)T+(−1.86e3−3.22e3i)T2 |
| 67 | 1+(−4.48+16.7i)T+(−3.88e3−2.24e3i)T2 |
| 71 | 1+(38.8−10.4i)T+(4.36e3−2.52e3i)T2 |
| 73 | 1+(−76.5+76.5i)T−5.32e3iT2 |
| 79 | 1+94.0T+6.24e3T2 |
| 83 | 1+(−33.6−33.6i)T+6.88e3iT2 |
| 89 | 1+(35.0+9.39i)T+(6.85e3+3.96e3i)T2 |
| 97 | 1+(−40.1+10.7i)T+(8.14e3−4.70e3i)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
−12.62714290854710913652195072672, −11.44461345564866401151999956673, −9.879309404989894431069582263308, −9.068326832257484902710796694367, −8.160358600818152860563275007953, −7.50460972570149879862929131429, −6.15310656577756503885213410726, −4.75801586329778902402611648251, −3.92641203491853923619597350671, −0.24261133922988061036500242751,
2.07533911828733906491818040184, 3.05425464923891770394525947691, 4.31901653940667426150837757282, 6.60506012672403692983224930280, 7.79862436917114440173097732412, 8.601609711895835011785117062058, 10.02810016541315238105503428933, 10.83731430973075783277902563943, 11.49616873307482845059035463016, 12.52723449886998485052934338405