L(s) = 1 | + (0.198 − 0.980i)2-s + (−0.0285 + 0.999i)3-s + (−0.921 − 0.389i)4-s + (−0.998 + 0.0570i)5-s + (0.974 + 0.226i)6-s + (0.897 − 0.441i)7-s + (−0.564 + 0.825i)8-s + (−0.998 − 0.0570i)9-s + (−0.142 + 0.989i)10-s + (0.415 − 0.909i)12-s + (0.696 − 0.717i)13-s + (−0.254 − 0.967i)14-s + (−0.0285 − 0.999i)15-s + (0.696 + 0.717i)16-s + (0.974 + 0.226i)17-s + (−0.254 + 0.967i)18-s + ⋯ |
L(s) = 1 | + (0.198 − 0.980i)2-s + (−0.0285 + 0.999i)3-s + (−0.921 − 0.389i)4-s + (−0.998 + 0.0570i)5-s + (0.974 + 0.226i)6-s + (0.897 − 0.441i)7-s + (−0.564 + 0.825i)8-s + (−0.998 − 0.0570i)9-s + (−0.142 + 0.989i)10-s + (0.415 − 0.909i)12-s + (0.696 − 0.717i)13-s + (−0.254 − 0.967i)14-s + (−0.0285 − 0.999i)15-s + (0.696 + 0.717i)16-s + (0.974 + 0.226i)17-s + (−0.254 + 0.967i)18-s + ⋯ |
Λ(s)=(=(253s/2ΓR(s)L(s)(0.841−0.540i)Λ(1−s)
Λ(s)=(=(253s/2ΓR(s)L(s)(0.841−0.540i)Λ(1−s)
Degree: |
1 |
Conductor: |
253
= 11⋅23
|
Sign: |
0.841−0.540i
|
Analytic conductor: |
1.17492 |
Root analytic conductor: |
1.17492 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ253(163,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 253, (0: ), 0.841−0.540i)
|
Particular Values
L(21) |
≈ |
1.062490158−0.3116498072i |
L(21) |
≈ |
1.062490158−0.3116498072i |
L(1) |
≈ |
0.9827611470−0.2352011888i |
L(1) |
≈ |
0.9827611470−0.2352011888i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 11 | 1 |
| 23 | 1 |
good | 2 | 1+(0.198−0.980i)T |
| 3 | 1+(−0.0285+0.999i)T |
| 5 | 1+(−0.998+0.0570i)T |
| 7 | 1+(0.897−0.441i)T |
| 13 | 1+(0.696−0.717i)T |
| 17 | 1+(0.974+0.226i)T |
| 19 | 1+(0.516+0.856i)T |
| 29 | 1+(0.516−0.856i)T |
| 31 | 1+(0.610+0.791i)T |
| 37 | 1+(0.774+0.633i)T |
| 41 | 1+(0.774−0.633i)T |
| 43 | 1+(−0.959+0.281i)T |
| 47 | 1+(0.309−0.951i)T |
| 53 | 1+(−0.466+0.884i)T |
| 59 | 1+(−0.985+0.170i)T |
| 61 | 1+(0.941−0.336i)T |
| 67 | 1+(0.415+0.909i)T |
| 71 | 1+(−0.870−0.491i)T |
| 73 | 1+(−0.921−0.389i)T |
| 79 | 1+(0.696−0.717i)T |
| 83 | 1+(−0.362+0.931i)T |
| 89 | 1+(−0.959+0.281i)T |
| 97 | 1+(−0.362−0.931i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−25.872438566676087901929628352150, −24.94514451202704895206457487761, −24.13507187418197753315153192672, −23.60450249541416874387644730391, −22.89932290806210403980327143541, −21.71352212729877841548647203458, −20.53103312111830781434243877091, −19.24851359903834750945273273198, −18.49991162109469606733436201708, −17.80694191065201971290510348455, −16.68038423474562740016324075549, −15.79908475570017701473233151007, −14.70381730452012130627068398243, −14.041093742503208553078801746207, −12.91276737509621492130349172685, −11.93698343331627655607827765780, −11.25313276880281350742413659179, −9.16278233411829954087735782710, −8.24864750045493040779579911115, −7.61926476537113082642474288229, −6.658775386888316618016614463918, −5.49105680575332229506841482296, −4.42512896796050184199263270853, −3.00824934981531131612434387471, −1.085486938796152465037120021926,
1.07466202351102894717573131829, 3.02668034010596618856052665843, 3.85128876899249156979585970432, 4.69885794119394237658332396416, 5.74873789310631815170522042371, 7.91586569851715843062941310313, 8.51362221591460590891288865856, 9.97616957022307347241045126244, 10.65938011127270250569411616974, 11.516739009518331708982405606541, 12.23958222218163431154495275898, 13.754683343576210260315326000767, 14.608381275687937030056379786156, 15.40837530948293317649278323162, 16.561534023296834367689888787623, 17.65212380396045201759577331713, 18.69574302159987114180460228137, 19.77851691443257760793379179929, 20.56273827124876587740278847549, 21.05137366258554635447868646522, 22.14589457563457259187220359642, 23.281371603801076380165579552028, 23.33931248753184675791122528761, 24.98916586539728329093360540379, 26.53196925024481015700774361640