L(s) = 1 | + (0.258 + 0.965i)2-s + (0.608 − 0.793i)3-s + (−0.866 + 0.5i)4-s + (0.608 − 0.793i)5-s + (0.923 + 0.382i)6-s + (0.130 + 0.991i)7-s + (−0.707 − 0.707i)8-s + (−0.258 − 0.965i)9-s + (0.923 + 0.382i)10-s + (−0.382 + 0.923i)11-s + (−0.130 + 0.991i)12-s + (0.5 + 0.866i)13-s + (−0.923 + 0.382i)14-s + (−0.258 − 0.965i)15-s + (0.5 − 0.866i)16-s + ⋯ |
L(s) = 1 | + (0.258 + 0.965i)2-s + (0.608 − 0.793i)3-s + (−0.866 + 0.5i)4-s + (0.608 − 0.793i)5-s + (0.923 + 0.382i)6-s + (0.130 + 0.991i)7-s + (−0.707 − 0.707i)8-s + (−0.258 − 0.965i)9-s + (0.923 + 0.382i)10-s + (−0.382 + 0.923i)11-s + (−0.130 + 0.991i)12-s + (0.5 + 0.866i)13-s + (−0.923 + 0.382i)14-s + (−0.258 − 0.965i)15-s + (0.5 − 0.866i)16-s + ⋯ |
Λ(s)=(=(629s/2ΓR(s)L(s)(0.555+0.831i)Λ(1−s)
Λ(s)=(=(629s/2ΓR(s)L(s)(0.555+0.831i)Λ(1−s)
Degree: |
1 |
Conductor: |
629
= 17⋅37
|
Sign: |
0.555+0.831i
|
Analytic conductor: |
2.92106 |
Root analytic conductor: |
2.92106 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ629(177,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 629, (0: ), 0.555+0.831i)
|
Particular Values
L(21) |
≈ |
1.774018666+0.9485879743i |
L(21) |
≈ |
1.774018666+0.9485879743i |
L(1) |
≈ |
1.414724563+0.4706000758i |
L(1) |
≈ |
1.414724563+0.4706000758i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 17 | 1 |
| 37 | 1 |
good | 2 | 1+(0.258+0.965i)T |
| 3 | 1+(0.608−0.793i)T |
| 5 | 1+(0.608−0.793i)T |
| 7 | 1+(0.130+0.991i)T |
| 11 | 1+(−0.382+0.923i)T |
| 13 | 1+(0.5+0.866i)T |
| 19 | 1+(0.965+0.258i)T |
| 23 | 1+(0.923+0.382i)T |
| 29 | 1+(0.382+0.923i)T |
| 31 | 1+(−0.923+0.382i)T |
| 41 | 1+(−0.130−0.991i)T |
| 43 | 1+(0.707+0.707i)T |
| 47 | 1−iT |
| 53 | 1+(0.965−0.258i)T |
| 59 | 1+(0.965−0.258i)T |
| 61 | 1+(0.991−0.130i)T |
| 67 | 1+(−0.5−0.866i)T |
| 71 | 1+(−0.608+0.793i)T |
| 73 | 1+(0.923−0.382i)T |
| 79 | 1+(0.130+0.991i)T |
| 83 | 1+(−0.965+0.258i)T |
| 89 | 1+(0.5−0.866i)T |
| 97 | 1+(−0.382−0.923i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−22.527443448111390109873249947550, −21.94983179852547684047356780726, −20.95197091679447229508244091927, −20.67050382925188252081899181006, −19.69602186736416074055588074497, −18.93201927855987220789548605770, −18.05582510033296647479196910106, −17.1864422699279172215219181837, −16.07805763860887142678829242055, −15.00355344729382886394654109675, −14.30593667812399267934642901587, −13.48503791945601303835166985756, −13.206280554490986845319398006550, −11.39529155052070488659447949692, −10.81902589699403142381415990783, −10.26012036614455721990941579883, −9.47830410042897363385394151455, −8.468431900343796113568067409755, −7.458891757794327183263357089222, −5.93101530497129816572240753693, −5.11565151496134931773647377436, −3.93113574092233242691074688332, −3.18431737456623314607458211059, −2.514072758810834030099090466330, −1.008402466918675634015541039910,
1.32784479897312355196523786835, 2.38246028313040204759599232043, 3.664864381448212762389047844498, 5.00496272519653770630313761546, 5.61786844659159989048050673479, 6.69737924749082820889154989261, 7.45617452706486760515523357179, 8.58292307368377627198531128129, 9.003756747940993542180093947657, 9.77561007790065209814899052066, 11.72751550610615928448513902876, 12.5323542872855277631174087814, 13.05086753341870769286741083561, 13.96754010828533090874068579009, 14.625454278697445697792892102782, 15.56326562040498988206029727045, 16.334688000753165871422230072309, 17.40294851748205952089666563426, 18.122906117096068549663937290024, 18.576598343500625441923959153813, 19.76233254864852616408490015626, 20.86598595972104113465962545562, 21.32914353213273304591489151028, 22.41740826705464543896263445082, 23.43373750093053224883369029002