L(s) = 1 | + (0.189 − 0.831i)2-s + (0.623 − 0.781i)3-s + (1.14 + 0.552i)4-s + (0.623 − 0.781i)5-s + (−0.531 − 0.666i)6-s + (−2.46 − 0.952i)7-s + (1.73 − 2.18i)8-s + (−0.222 − 0.974i)9-s + (−0.531 − 0.666i)10-s + (−0.625 + 2.74i)11-s + (1.14 − 0.552i)12-s + (1.23 − 5.42i)13-s + (−1.26 + 1.87i)14-s + (−0.222 − 0.974i)15-s + (0.104 + 0.131i)16-s + (3.87 − 1.86i)17-s + ⋯ |
L(s) = 1 | + (0.134 − 0.587i)2-s + (0.359 − 0.451i)3-s + (0.573 + 0.276i)4-s + (0.278 − 0.349i)5-s + (−0.216 − 0.272i)6-s + (−0.932 − 0.360i)7-s + (0.615 − 0.771i)8-s + (−0.0741 − 0.324i)9-s + (−0.168 − 0.210i)10-s + (−0.188 + 0.826i)11-s + (0.331 − 0.159i)12-s + (0.343 − 1.50i)13-s + (−0.336 + 0.499i)14-s + (−0.0574 − 0.251i)15-s + (0.0262 + 0.0329i)16-s + (0.939 − 0.452i)17-s + ⋯ |
Λ(s)=(=(735s/2ΓC(s)L(s)(−0.253+0.967i)Λ(2−s)
Λ(s)=(=(735s/2ΓC(s+1/2)L(s)(−0.253+0.967i)Λ(1−s)
Degree: |
2 |
Conductor: |
735
= 3⋅5⋅72
|
Sign: |
−0.253+0.967i
|
Analytic conductor: |
5.86900 |
Root analytic conductor: |
2.42260 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ735(211,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 735, ( :1/2), −0.253+0.967i)
|
Particular Values
L(1) |
≈ |
1.27190−1.64776i |
L(21) |
≈ |
1.27190−1.64776i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(−0.623+0.781i)T |
| 5 | 1+(−0.623+0.781i)T |
| 7 | 1+(2.46+0.952i)T |
good | 2 | 1+(−0.189+0.831i)T+(−1.80−0.867i)T2 |
| 11 | 1+(0.625−2.74i)T+(−9.91−4.77i)T2 |
| 13 | 1+(−1.23+5.42i)T+(−11.7−5.64i)T2 |
| 17 | 1+(−3.87+1.86i)T+(10.5−13.2i)T2 |
| 19 | 1+1.56T+19T2 |
| 23 | 1+(5.71+2.75i)T+(14.3+17.9i)T2 |
| 29 | 1+(2.35−1.13i)T+(18.0−22.6i)T2 |
| 31 | 1−8.87T+31T2 |
| 37 | 1+(−4.72+2.27i)T+(23.0−28.9i)T2 |
| 41 | 1+(1.73−2.17i)T+(−9.12−39.9i)T2 |
| 43 | 1+(−2.68−3.36i)T+(−9.56+41.9i)T2 |
| 47 | 1+(2.58−11.3i)T+(−42.3−20.3i)T2 |
| 53 | 1+(0.879+0.423i)T+(33.0+41.4i)T2 |
| 59 | 1+(3.43+4.31i)T+(−13.1+57.5i)T2 |
| 61 | 1+(−10.8+5.24i)T+(38.0−47.6i)T2 |
| 67 | 1+8.36T+67T2 |
| 71 | 1+(4.72+2.27i)T+(44.2+55.5i)T2 |
| 73 | 1+(−2.77−12.1i)T+(−65.7+31.6i)T2 |
| 79 | 1+5.88T+79T2 |
| 83 | 1+(−1.75−7.68i)T+(−74.7+36.0i)T2 |
| 89 | 1+(−0.778−3.41i)T+(−80.1+38.6i)T2 |
| 97 | 1+6.73T+97T2 |
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
−10.02350533997243749660197059186, −9.693735448004402854660477475604, −8.162225216175250773152090144661, −7.66796769043208589847850381642, −6.63135071073122320509805375356, −5.84472492984896821317880261178, −4.35478732329581819172033272306, −3.22884404606525913981400223254, −2.47774080176655663271990301530, −1.01611333791507798116387747777,
1.94013692635664205687647030201, 3.08000928147509741993829871313, 4.17263051863819818109577888555, 5.64464759050625587358327558849, 6.14572863255748686633224979245, 6.95124254242505271649228010488, 8.037773172549761441239133036335, 8.870405830364642140702078734342, 9.903055260266149774441928372006, 10.37790443977387496623566460001