L(s) = 1 | + (0.529 + 0.848i)2-s + (−0.438 + 0.898i)4-s + (−0.984 − 0.173i)7-s + (−0.994 + 0.104i)8-s + (−0.0348 − 0.999i)11-s + (−0.529 + 0.848i)13-s + (−0.374 − 0.927i)14-s + (−0.615 − 0.788i)16-s + (−0.406 + 0.913i)17-s + (0.104 + 0.994i)19-s + (0.829 − 0.559i)22-s + (0.139 − 0.990i)23-s − 26-s + (0.587 − 0.809i)28-s + (−0.241 − 0.970i)29-s + ⋯ |
L(s) = 1 | + (0.529 + 0.848i)2-s + (−0.438 + 0.898i)4-s + (−0.984 − 0.173i)7-s + (−0.994 + 0.104i)8-s + (−0.0348 − 0.999i)11-s + (−0.529 + 0.848i)13-s + (−0.374 − 0.927i)14-s + (−0.615 − 0.788i)16-s + (−0.406 + 0.913i)17-s + (0.104 + 0.994i)19-s + (0.829 − 0.559i)22-s + (0.139 − 0.990i)23-s − 26-s + (0.587 − 0.809i)28-s + (−0.241 − 0.970i)29-s + ⋯ |
Λ(s)=(=(675s/2ΓR(s)L(s)(0.00930−0.999i)Λ(1−s)
Λ(s)=(=(675s/2ΓR(s)L(s)(0.00930−0.999i)Λ(1−s)
Degree: |
1 |
Conductor: |
675
= 33⋅52
|
Sign: |
0.00930−0.999i
|
Analytic conductor: |
3.13468 |
Root analytic conductor: |
3.13468 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ675(23,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 675, (0: ), 0.00930−0.999i)
|
Particular Values
L(21) |
≈ |
0.1282428649−0.1270546478i |
L(21) |
≈ |
0.1282428649−0.1270546478i |
L(1) |
≈ |
0.7639118757+0.3577900787i |
L(1) |
≈ |
0.7639118757+0.3577900787i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 5 | 1 |
good | 2 | 1+(0.529+0.848i)T |
| 7 | 1+(−0.984−0.173i)T |
| 11 | 1+(−0.0348−0.999i)T |
| 13 | 1+(−0.529+0.848i)T |
| 17 | 1+(−0.406+0.913i)T |
| 19 | 1+(0.104+0.994i)T |
| 23 | 1+(0.139−0.990i)T |
| 29 | 1+(−0.241−0.970i)T |
| 31 | 1+(−0.719−0.694i)T |
| 37 | 1+(−0.743−0.669i)T |
| 41 | 1+(−0.848−0.529i)T |
| 43 | 1+(−0.342−0.939i)T |
| 47 | 1+(−0.694−0.719i)T |
| 53 | 1+(−0.587+0.809i)T |
| 59 | 1+(0.0348−0.999i)T |
| 61 | 1+(−0.882−0.469i)T |
| 67 | 1+(0.970+0.241i)T |
| 71 | 1+(0.104−0.994i)T |
| 73 | 1+(−0.743+0.669i)T |
| 79 | 1+(0.241+0.970i)T |
| 83 | 1+(−0.0697+0.997i)T |
| 89 | 1+(0.669+0.743i)T |
| 97 | 1+(0.275−0.961i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−22.74235045015576027597638505814, −22.1961791988889169392937787219, −21.46603570782534809398694960260, −20.16051195001202459708565213678, −20.04681996510013441574707174632, −19.1220021330213394481690307579, −18.09250436021077649983838868492, −17.54789744877373118478234315394, −16.12494827006930737101009492149, −15.361926640019587946760445794357, −14.67090657124199754268914422463, −13.473651260816980243704904790369, −12.952930317184261688257366959352, −12.188555842110710557352696920519, −11.330512536092546058219135768283, −10.28223287591958839252349156353, −9.61621434952439398926204546893, −8.927717833829796014241440302021, −7.34587758393836083553910886810, −6.55551453841262875441658126685, −5.28891936562399101986530640491, −4.73150751217133362594985706116, −3.32740400975208251125932181457, −2.79192722375557864904222945263, −1.543112721931446707372746981677,
0.069380423234969512169798230553, 2.21169230323186325970743718374, 3.50840911398879826674975774246, 4.0575185163800822600828036838, 5.363035699901945329455984928396, 6.23860449184586505378287987515, 6.82743888261567445532060500523, 7.94629472383583606733294190692, 8.78718899778069848368506881163, 9.66332172561171784846692857502, 10.78318206617136458273028983072, 11.97724352860337499397966672669, 12.687229187205579565977679528639, 13.546213479600863397036108986057, 14.215259562823025714033333256585, 15.13353361203868027259839696755, 16.00464206559009499893786539098, 16.76061096036969763952195826713, 17.13037046019835190280991915737, 18.626152350934431098742864020423, 19.00052940697207612066005996086, 20.19465387751861112107043019337, 21.21594220934585685286740737566, 21.95369404168799654951420409181, 22.55790008203230475629940897768