L(s) = 1 | + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.913 − 0.406i)5-s + (−0.309 + 0.951i)8-s + (−0.5 − 0.866i)10-s + (0.913 − 0.406i)13-s + (−0.809 + 0.587i)16-s + (−0.913 − 0.406i)17-s + (0.669 + 0.743i)19-s + (0.104 − 0.994i)20-s + (0.5 + 0.866i)23-s + (0.669 + 0.743i)25-s + (0.978 + 0.207i)26-s + (−0.669 + 0.743i)29-s + (−0.809 − 0.587i)31-s − 32-s + ⋯ |
L(s) = 1 | + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.913 − 0.406i)5-s + (−0.309 + 0.951i)8-s + (−0.5 − 0.866i)10-s + (0.913 − 0.406i)13-s + (−0.809 + 0.587i)16-s + (−0.913 − 0.406i)17-s + (0.669 + 0.743i)19-s + (0.104 − 0.994i)20-s + (0.5 + 0.866i)23-s + (0.669 + 0.743i)25-s + (0.978 + 0.207i)26-s + (−0.669 + 0.743i)29-s + (−0.809 − 0.587i)31-s − 32-s + ⋯ |
Λ(s)=(=(693s/2ΓR(s+1)L(s)(−0.292−0.956i)Λ(1−s)
Λ(s)=(=(693s/2ΓR(s+1)L(s)(−0.292−0.956i)Λ(1−s)
Degree: |
1 |
Conductor: |
693
= 32⋅7⋅11
|
Sign: |
−0.292−0.956i
|
Analytic conductor: |
74.4731 |
Root analytic conductor: |
74.4731 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ693(515,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 693, (1: ), −0.292−0.956i)
|
Particular Values
L(21) |
≈ |
0.1776683798−0.2401457858i |
L(21) |
≈ |
0.1776683798−0.2401457858i |
L(1) |
≈ |
1.102696826+0.3859584811i |
L(1) |
≈ |
1.102696826+0.3859584811i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1 |
| 11 | 1 |
good | 2 | 1+(0.809+0.587i)T |
| 5 | 1+(−0.913−0.406i)T |
| 13 | 1+(0.913−0.406i)T |
| 17 | 1+(−0.913−0.406i)T |
| 19 | 1+(0.669+0.743i)T |
| 23 | 1+(0.5+0.866i)T |
| 29 | 1+(−0.669+0.743i)T |
| 31 | 1+(−0.809−0.587i)T |
| 37 | 1+(−0.978−0.207i)T |
| 41 | 1+(−0.669−0.743i)T |
| 43 | 1+(−0.5−0.866i)T |
| 47 | 1+(−0.309+0.951i)T |
| 53 | 1+(0.104−0.994i)T |
| 59 | 1+(−0.309−0.951i)T |
| 61 | 1+(−0.809+0.587i)T |
| 67 | 1+T |
| 71 | 1+(0.809−0.587i)T |
| 73 | 1+(0.669−0.743i)T |
| 79 | 1+(−0.809−0.587i)T |
| 83 | 1+(−0.913−0.406i)T |
| 89 | 1+(0.5−0.866i)T |
| 97 | 1+(−0.104+0.994i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−22.76791907870489526302961723513, −21.9497826065951902902700557827, −21.18520341791217421646871365252, −20.11910622432803916092780578300, −19.7932074123603782200089365312, −18.67172781741168357336128989783, −18.288019365657169115699650491, −16.79392184045739536211646262046, −15.747432393171253407942683144545, −15.306399532003701606536101573802, −14.39412892764934442880711012431, −13.51518613955657744094127035364, −12.76099006770254475856783034263, −11.737555337346956292169856095893, −11.163698112677961319956961971017, −10.51915495142643927029700701796, −9.278551073421319326939583640795, −8.34024268106344307890634101659, −7.00396210979749942393843162655, −6.45731622801961386490989473636, −5.180384952435514767507244238293, −4.2470463026254763410044078954, −3.5025420083424202417334839883, −2.54457808998680144333159263289, −1.27839409807207971673802284971,
0.05148222250034412616235393522, 1.702387595898033167543379502747, 3.32351515889929410447284803036, 3.76804180876861193495933096285, 4.94937380341285469316748157190, 5.63357318110930093521932452293, 6.85406368826913655846764563200, 7.57674122268196415417176382198, 8.44043334649705840395794744893, 9.22054147654271470695931382069, 10.86855889524570617570518586748, 11.491726373269862205753390368582, 12.378996598376016262461010508888, 13.14626058120251198766047256339, 13.89741350818404238945418852251, 14.97041761184928592879861349283, 15.65145235119978053583325077053, 16.19245337831187384444185279174, 17.06461071237603093544459249763, 18.03325753254729412040742294874, 18.95881209440797909257705610929, 20.28190029017798997754258386791, 20.43348249172407834261639745098, 21.5331120860216442252766529132, 22.58944734650220453903558889873