L(s) = 1 | + (0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + 6-s + (−0.866 − 0.5i)7-s − i·8-s + (0.5 + 0.866i)9-s + 11-s + (0.866 − 0.5i)12-s + (0.866 + 0.5i)13-s − 14-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s + (0.866 + 0.5i)18-s + (0.5 − 0.866i)19-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + 6-s + (−0.866 − 0.5i)7-s − i·8-s + (0.5 + 0.866i)9-s + 11-s + (0.866 − 0.5i)12-s + (0.866 + 0.5i)13-s − 14-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s + (0.866 + 0.5i)18-s + (0.5 − 0.866i)19-s + ⋯ |
Λ(s)=(=(185s/2ΓR(s+1)L(s)(0.644−0.764i)Λ(1−s)
Λ(s)=(=(185s/2ΓR(s+1)L(s)(0.644−0.764i)Λ(1−s)
Degree: |
1 |
Conductor: |
185
= 5⋅37
|
Sign: |
0.644−0.764i
|
Analytic conductor: |
19.8810 |
Root analytic conductor: |
19.8810 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ185(47,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 185, (1: ), 0.644−0.764i)
|
Particular Values
L(21) |
≈ |
3.875992115−1.801404158i |
L(21) |
≈ |
3.875992115−1.801404158i |
L(1) |
≈ |
2.272492680−0.6408059560i |
L(1) |
≈ |
2.272492680−0.6408059560i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1 |
| 37 | 1 |
good | 2 | 1+(0.866−0.5i)T |
| 3 | 1+(0.866+0.5i)T |
| 7 | 1+(−0.866−0.5i)T |
| 11 | 1+T |
| 13 | 1+(0.866+0.5i)T |
| 17 | 1+(0.866−0.5i)T |
| 19 | 1+(0.5−0.866i)T |
| 23 | 1−iT |
| 29 | 1−T |
| 31 | 1+T |
| 41 | 1+(−0.5+0.866i)T |
| 43 | 1−iT |
| 47 | 1+iT |
| 53 | 1+(−0.866+0.5i)T |
| 59 | 1+(0.5+0.866i)T |
| 61 | 1+(−0.5+0.866i)T |
| 67 | 1+(−0.866−0.5i)T |
| 71 | 1+(−0.5+0.866i)T |
| 73 | 1−iT |
| 79 | 1+(0.5−0.866i)T |
| 83 | 1+(−0.866+0.5i)T |
| 89 | 1+(0.5+0.866i)T |
| 97 | 1+iT |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−26.72082975511269388788276523433, −25.61795081194035479807986037581, −25.303106059733753203150855030530, −24.41177815890740547204473743702, −23.28905461877097229934251683314, −22.510551196936095943057988204723, −21.397588002544612846044779263856, −20.48730816343648327731117840560, −19.50224155168371676819341327674, −18.54092253368550393821695231752, −17.2309593898328085161842863875, −16.06209896741638570017752220385, −15.19603730965151302936457658557, −14.28795454086290985936878119973, −13.37659127331363198395426256591, −12.53748475326376151029584332346, −11.67700407535228486058130936984, −9.8056813014266473185788402915, −8.65229193918472378844959175673, −7.67715961773922533101882710214, −6.50177132890227714893185318017, −5.70046445253553867008226375149, −3.738541196627590531902666385335, −3.24072786253104252002218783458, −1.61993516509753482632350990938,
1.20134797481881114648601274575, 2.82049594341687468815554555004, 3.69515324847073128294684559911, 4.608193522282537551929049323528, 6.19622229551611807440014522972, 7.26060744371695802824067612867, 9.01383298558352160423664736529, 9.797470919408957046045482185658, 10.84541208231910676706980063454, 12.02682165581028253916214873626, 13.31308705604730998822071712718, 13.89895896128479385301374689105, 14.8301941283994161144045961130, 15.92988545111724298012316185896, 16.64911719043803901919544880160, 18.68370945299884975111557083407, 19.41745997611607047823260453202, 20.299211503241279928607447108023, 20.95402642002816193238180711624, 22.12873053819101222180299920903, 22.704699135636709030392380340805, 23.90868652277994579812713154829, 24.96705955159777559605389967001, 25.7754978819631957054914622379, 26.79946702661941151262149677047