L(s) = 1 | + (−0.473 − 0.880i)2-s + (−0.550 − 0.834i)3-s + (−0.550 + 0.834i)4-s + (0.983 − 0.178i)5-s + (−0.473 + 0.880i)6-s + (−0.936 − 0.351i)7-s + (0.995 + 0.0896i)8-s + (−0.393 + 0.919i)9-s + (−0.623 − 0.781i)10-s + 12-s + (−0.753 + 0.657i)13-s + (0.134 + 0.990i)14-s + (−0.691 − 0.722i)15-s + (−0.393 − 0.919i)16-s + (0.809 − 0.587i)17-s + (0.995 − 0.0896i)18-s + ⋯ |
L(s) = 1 | + (−0.473 − 0.880i)2-s + (−0.550 − 0.834i)3-s + (−0.550 + 0.834i)4-s + (0.983 − 0.178i)5-s + (−0.473 + 0.880i)6-s + (−0.936 − 0.351i)7-s + (0.995 + 0.0896i)8-s + (−0.393 + 0.919i)9-s + (−0.623 − 0.781i)10-s + 12-s + (−0.753 + 0.657i)13-s + (0.134 + 0.990i)14-s + (−0.691 − 0.722i)15-s + (−0.393 − 0.919i)16-s + (0.809 − 0.587i)17-s + (0.995 − 0.0896i)18-s + ⋯ |
Λ(s)=(=(319s/2ΓR(s+1)L(s)(0.417−0.908i)Λ(1−s)
Λ(s)=(=(319s/2ΓR(s+1)L(s)(0.417−0.908i)Λ(1−s)
Degree: |
1 |
Conductor: |
319
= 11⋅29
|
Sign: |
0.417−0.908i
|
Analytic conductor: |
34.2813 |
Root analytic conductor: |
34.2813 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ319(112,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 319, (1: ), 0.417−0.908i)
|
Particular Values
L(21) |
≈ |
0.8226586349−0.5273578598i |
L(21) |
≈ |
0.8226586349−0.5273578598i |
L(1) |
≈ |
0.5945074209−0.3748507076i |
L(1) |
≈ |
0.5945074209−0.3748507076i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 11 | 1 |
| 29 | 1 |
good | 2 | 1+(−0.473−0.880i)T |
| 3 | 1+(−0.550−0.834i)T |
| 5 | 1+(0.983−0.178i)T |
| 7 | 1+(−0.936−0.351i)T |
| 13 | 1+(−0.753+0.657i)T |
| 17 | 1+(0.809−0.587i)T |
| 19 | 1+(−0.936+0.351i)T |
| 23 | 1+(−0.900+0.433i)T |
| 31 | 1+(0.473+0.880i)T |
| 37 | 1+(0.858−0.512i)T |
| 41 | 1+(−0.309+0.951i)T |
| 43 | 1+(0.900−0.433i)T |
| 47 | 1+(0.858+0.512i)T |
| 53 | 1+(0.473+0.880i)T |
| 59 | 1+(0.309+0.951i)T |
| 61 | 1+(0.0448−0.998i)T |
| 67 | 1+(−0.222−0.974i)T |
| 71 | 1+(−0.393−0.919i)T |
| 73 | 1+(0.691+0.722i)T |
| 79 | 1+(0.393−0.919i)T |
| 83 | 1+(0.963+0.266i)T |
| 89 | 1+(−0.900−0.433i)T |
| 97 | 1+(−0.0448−0.998i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−25.44629564214231184286343051952, −24.30369842767444653481477472835, −23.276090131328638739604952638120, −22.291945613536395580859355161850, −21.97734824555659239552025089849, −20.732242346888745237002276521788, −19.520377322277551464385331595210, −18.555270647823050012677578756136, −17.54469248729088681977718596311, −16.97170903605609545151331834666, −16.19280807431269036162726031806, −15.15996346022741787617673325763, −14.58681405910548271828443743503, −13.31266418491100620118583916496, −12.30689407700036543445340228272, −10.65514004056142362198105556658, −10.00997773876772264823856410945, −9.446168689915077504438333762878, −8.33439984287540955825619965979, −6.804214395406316419686630491734, −5.97505125072136105985470211816, −5.404941926712173162687724451385, −4.08571204654439173725755583275, −2.48903318784769837773844736647, −0.54921235985428212247746626774,
0.7478863562441804020204789011, 1.90270322263097189246966862717, 2.85190852777865354093087788527, 4.44590289704954460911746582533, 5.75758764303471875660061855047, 6.80501848227620061743924667278, 7.80433239505084363490439645263, 9.14820282056464673649356154658, 9.95528890291513905596958309009, 10.75730662168591534771857745285, 12.09765593504276752335366056185, 12.55508321491411779539916871250, 13.56541461294498402918639730708, 14.15585967893120813188731698281, 16.35779198240461078209771913237, 16.85205914827981101986950259303, 17.669020844697776698926561060429, 18.55318041300454534331840953742, 19.301682948720463169864502791673, 20.07718604347104655105497929979, 21.29017512747549016069865612146, 21.98372446752158296935724343532, 22.84841212127843798051716440449, 23.71458228675677402529819187318, 25.06769632145645807549375957134