L(s) = 1 | + (0.913 − 0.406i)2-s + (0.669 − 0.743i)4-s + (−0.809 + 0.587i)5-s + (0.309 − 0.951i)8-s + (−0.5 + 0.866i)10-s + (0.913 − 0.406i)13-s + (−0.104 − 0.994i)16-s + (−0.104 − 0.994i)17-s + (−0.978 + 0.207i)19-s + (−0.104 + 0.994i)20-s + 23-s + (0.309 − 0.951i)25-s + (0.669 − 0.743i)26-s + (0.669 − 0.743i)29-s + (−0.104 + 0.994i)31-s + (−0.5 − 0.866i)32-s + ⋯ |
L(s) = 1 | + (0.913 − 0.406i)2-s + (0.669 − 0.743i)4-s + (−0.809 + 0.587i)5-s + (0.309 − 0.951i)8-s + (−0.5 + 0.866i)10-s + (0.913 − 0.406i)13-s + (−0.104 − 0.994i)16-s + (−0.104 − 0.994i)17-s + (−0.978 + 0.207i)19-s + (−0.104 + 0.994i)20-s + 23-s + (0.309 − 0.951i)25-s + (0.669 − 0.743i)26-s + (0.669 − 0.743i)29-s + (−0.104 + 0.994i)31-s + (−0.5 − 0.866i)32-s + ⋯ |
Λ(s)=(=(693s/2ΓR(s)L(s)(0.346−0.938i)Λ(1−s)
Λ(s)=(=(693s/2ΓR(s)L(s)(0.346−0.938i)Λ(1−s)
Degree: |
1 |
Conductor: |
693
= 32⋅7⋅11
|
Sign: |
0.346−0.938i
|
Analytic conductor: |
3.21827 |
Root analytic conductor: |
3.21827 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ693(394,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 693, (0: ), 0.346−0.938i)
|
Particular Values
L(21) |
≈ |
1.798070396−1.253116154i |
L(21) |
≈ |
1.798070396−1.253116154i |
L(1) |
≈ |
1.543092972−0.5279193140i |
L(1) |
≈ |
1.543092972−0.5279193140i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1 |
| 11 | 1 |
good | 2 | 1+(0.913−0.406i)T |
| 5 | 1+(−0.809+0.587i)T |
| 13 | 1+(0.913−0.406i)T |
| 17 | 1+(−0.104−0.994i)T |
| 19 | 1+(−0.978+0.207i)T |
| 23 | 1+T |
| 29 | 1+(0.669−0.743i)T |
| 31 | 1+(−0.104+0.994i)T |
| 37 | 1+(0.669−0.743i)T |
| 41 | 1+(0.669+0.743i)T |
| 43 | 1+(−0.5−0.866i)T |
| 47 | 1+(0.669+0.743i)T |
| 53 | 1+(0.913−0.406i)T |
| 59 | 1+(0.669−0.743i)T |
| 61 | 1+(−0.104−0.994i)T |
| 67 | 1+(−0.5−0.866i)T |
| 71 | 1+(−0.809+0.587i)T |
| 73 | 1+(−0.978−0.207i)T |
| 79 | 1+(0.913−0.406i)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
−23.12559089301092915541532927272, −22.064776636365187090232168771774, −21.225287507021481840261154578278, −20.63250207530066797332277269, −19.698459344569216554893614208684, −18.985113833103469036898740327536, −17.69151144214389718394695106884, −16.70489388427980322310774790201, −16.2994277976597373103474191845, −15.1540630524389482543493664080, −14.915106847941956127593198356711, −13.52278693745110886585846528594, −12.99879044353149365878895419316, −12.15992230994328202613706970107, −11.32068502934993740735485415229, −10.598891154491219078943894140273, −8.86509387199948450001872631387, −8.39304534066193390704413085534, −7.34993605492453884135116661166, −6.45287873952278115566327094077, −5.533821017190706615705549358991, −4.385805605372028820683681878582, −3.94627464987155958879101847368, −2.76373388936548926372758673642, −1.37103172288662011057920857659,
0.864080134093522788289704962515, 2.39062157668879141888766140463, 3.23418105629946379628748992163, 4.0858800387077225230930390819, 4.9929575850710368421710443228, 6.16380233119397515743286538417, 6.89760624085658567676492003580, 7.86678080486136277437633249317, 9.02357609419326959342145755514, 10.3189449403534269721011460060, 10.95135455024735313136126781367, 11.64133894563765129937530498380, 12.528229849358372255011644073135, 13.36262440899573179664737856753, 14.25526538199397217474653856109, 15.015266430640771421424404827424, 15.73407595989786376509816319216, 16.393947101317211577751244396254, 17.81370072070080822535193829318, 18.736037753585776819532106617017, 19.34550472234071180192083927530, 20.1851761784255327012491615181, 20.95558513150632196001341658815, 21.7272632713930911943914336104, 22.73674878011173790109328753169