L(s) = 1 | + (0.835 + 0.549i)2-s + (0.396 + 0.918i)4-s + (0.286 − 0.957i)5-s + (−0.597 − 0.802i)7-s + (−0.173 + 0.984i)8-s + (0.766 − 0.642i)10-s + (0.286 + 0.957i)11-s + (−0.0581 − 0.998i)14-s + (−0.686 + 0.727i)16-s + (0.766 − 0.642i)17-s + (−0.173 + 0.984i)19-s + (0.993 − 0.116i)20-s + (−0.286 + 0.957i)22-s + (0.597 − 0.802i)23-s + (−0.835 − 0.549i)25-s + ⋯ |
L(s) = 1 | + (0.835 + 0.549i)2-s + (0.396 + 0.918i)4-s + (0.286 − 0.957i)5-s + (−0.597 − 0.802i)7-s + (−0.173 + 0.984i)8-s + (0.766 − 0.642i)10-s + (0.286 + 0.957i)11-s + (−0.0581 − 0.998i)14-s + (−0.686 + 0.727i)16-s + (0.766 − 0.642i)17-s + (−0.173 + 0.984i)19-s + (0.993 − 0.116i)20-s + (−0.286 + 0.957i)22-s + (0.597 − 0.802i)23-s + (−0.835 − 0.549i)25-s + ⋯ |
Λ(s)=(=(1053s/2ΓR(s)L(s)(0.928+0.372i)Λ(1−s)
Λ(s)=(=(1053s/2ΓR(s)L(s)(0.928+0.372i)Λ(1−s)
Degree: |
1 |
Conductor: |
1053
= 34⋅13
|
Sign: |
0.928+0.372i
|
Analytic conductor: |
4.89011 |
Root analytic conductor: |
4.89011 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1053(238,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 1053, (0: ), 0.928+0.372i)
|
Particular Values
L(21) |
≈ |
2.525841097+0.4874486014i |
L(21) |
≈ |
2.525841097+0.4874486014i |
L(1) |
≈ |
1.718117309+0.3207937665i |
L(1) |
≈ |
1.718117309+0.3207937665i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 13 | 1 |
good | 2 | 1+(0.835+0.549i)T |
| 5 | 1+(0.286−0.957i)T |
| 7 | 1+(−0.597−0.802i)T |
| 11 | 1+(0.286+0.957i)T |
| 17 | 1+(0.766−0.642i)T |
| 19 | 1+(−0.173+0.984i)T |
| 23 | 1+(0.597−0.802i)T |
| 29 | 1+(0.893−0.448i)T |
| 31 | 1+(0.993+0.116i)T |
| 37 | 1+(0.939+0.342i)T |
| 41 | 1+(0.0581+0.998i)T |
| 43 | 1+(−0.286−0.957i)T |
| 47 | 1+(0.993−0.116i)T |
| 53 | 1+(−0.5−0.866i)T |
| 59 | 1+(0.286−0.957i)T |
| 61 | 1+(−0.993+0.116i)T |
| 67 | 1+(0.835−0.549i)T |
| 71 | 1+(0.939+0.342i)T |
| 73 | 1+(−0.173+0.984i)T |
| 79 | 1+(0.893−0.448i)T |
| 83 | 1+(−0.893+0.448i)T |
| 89 | 1+(0.939−0.342i)T |
| 97 | 1+(−0.973−0.230i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−21.67390598709424377558861267537, −21.09528589598417896844742658655, −19.703699301157259229425006003246, −19.25398599307435878124861965933, −18.70427936201941065291841274880, −17.785783511776343488978202261141, −16.66448577327147756515235217645, −15.59625794689280897414075362697, −15.178953524440634419755522773760, −14.23264643875228311766390486918, −13.639374762440377986297862800630, −12.80225533156443904123320893269, −11.93721332411664468408151872238, −11.19505373542122117339545788247, −10.49994054913024059866706324554, −9.62471453957274016388614816170, −8.83639214561929754302229424762, −7.44558131651180090597528175155, −6.3507614021180708659565790452, −6.033663646211443973868505949914, −5.05809127232933787482082681812, −3.76321388911063762979416788528, −3.01284029248120937230195891854, −2.45206915957581111478291903888, −1.1008313145829288635774694668,
1.01239637071745075526012927585, 2.32845659961311490792029483044, 3.46494343446765923813807134147, 4.41718862952213921092936391049, 4.92598639250100152960482605271, 6.06171981125898719908758910365, 6.75174781603647924172675498924, 7.696551253438326548123374667454, 8.43717525922726625729208346724, 9.60113122518278251440495283346, 10.21426414259401342542273992329, 11.5793221771913667743164836545, 12.430602076830698038189969870088, 12.80480508081589209958273161184, 13.78649158283666987322895543404, 14.29504923921563432037134180203, 15.34815129434049484599134350010, 16.13040200530655701223640194510, 16.89525988854754120056852488937, 17.152442301953051144334694222527, 18.28753080740992211540948846183, 19.502570345593238975751097312512, 20.421858620341867825536467366302, 20.68563326449464932123683169384, 21.57281090663734858068537397132