L(s) = 1 | + (0.866 − 0.5i)2-s + (0.158 − 1.72i)3-s + (0.499 − 0.866i)4-s + (−0.917 + 2.03i)5-s + (−0.724 − 1.57i)6-s + (0.389 − 0.224i)7-s − 0.999i·8-s + (−2.94 − 0.548i)9-s + (0.224 + 2.22i)10-s + (1.72 + 2.98i)11-s + (−1.41 − 0.999i)12-s + (−2.12 − 1.22i)13-s + (0.224 − 0.389i)14-s + (3.37 + 1.90i)15-s + (−0.5 − 0.866i)16-s + 5.89i·17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.0917 − 0.995i)3-s + (0.249 − 0.433i)4-s + (−0.410 + 0.911i)5-s + (−0.295 − 0.642i)6-s + (0.147 − 0.0849i)7-s − 0.353i·8-s + (−0.983 − 0.182i)9-s + (0.0710 + 0.703i)10-s + (0.520 + 0.900i)11-s + (−0.408 − 0.288i)12-s + (−0.588 − 0.339i)13-s + (0.0600 − 0.104i)14-s + (0.870 + 0.492i)15-s + (−0.125 − 0.216i)16-s + 1.43i·17-s + ⋯ |
Λ(s)=(=(90s/2ΓC(s)L(s)(0.576+0.816i)Λ(2−s)
Λ(s)=(=(90s/2ΓC(s+1/2)L(s)(0.576+0.816i)Λ(1−s)
Degree: |
2 |
Conductor: |
90
= 2⋅32⋅5
|
Sign: |
0.576+0.816i
|
Analytic conductor: |
0.718653 |
Root analytic conductor: |
0.847734 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ90(49,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 90, ( :1/2), 0.576+0.816i)
|
Particular Values
L(1) |
≈ |
1.12926−0.584859i |
L(21) |
≈ |
1.12926−0.584859i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.866+0.5i)T |
| 3 | 1+(−0.158+1.72i)T |
| 5 | 1+(0.917−2.03i)T |
good | 7 | 1+(−0.389+0.224i)T+(3.5−6.06i)T2 |
| 11 | 1+(−1.72−2.98i)T+(−5.5+9.52i)T2 |
| 13 | 1+(2.12+1.22i)T+(6.5+11.2i)T2 |
| 17 | 1−5.89iT−17T2 |
| 19 | 1−5.44T+19T2 |
| 23 | 1+(5.97+3.44i)T+(11.5+19.9i)T2 |
| 29 | 1+(3+5.19i)T+(−14.5+25.1i)T2 |
| 31 | 1+(0.775−1.34i)T+(−15.5−26.8i)T2 |
| 37 | 1+8iT−37T2 |
| 41 | 1+(−0.5+0.866i)T+(−20.5−35.5i)T2 |
| 43 | 1+(−2.20+1.27i)T+(21.5−37.2i)T2 |
| 47 | 1+(−3.85+2.22i)T+(23.5−40.7i)T2 |
| 53 | 1−3.55iT−53T2 |
| 59 | 1+(−6.62+11.4i)T+(−29.5−51.0i)T2 |
| 61 | 1+(2.22+3.85i)T+(−30.5+52.8i)T2 |
| 67 | 1+(−3.94−2.27i)T+(33.5+58.0i)T2 |
| 71 | 1+2.44T+71T2 |
| 73 | 1−14.7iT−73T2 |
| 79 | 1+(−3.67−6.36i)T+(−39.5+68.4i)T2 |
| 83 | 1+(3.46−2i)T+(41.5−71.8i)T2 |
| 89 | 1+3.10T+89T2 |
| 97 | 1+(11.2−6.5i)T+(48.5−84.0i)T2 |
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L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−14.10785871550535578383461112336, −12.70388573077722447510903210273, −12.06436702077582165151651079696, −11.03608042251989246659065417683, −9.823977150577299661080253324953, −7.964055672745042024316698617247, −7.03359102898768566334571652894, −5.86777420491565887386810951752, −3.87973663604563963223802631918, −2.22143501198063388053769767323,
3.40143802318796413810831655341, 4.71292075716634527105090508002, 5.64602718897363475251973089842, 7.55237684885516684005224754985, 8.804359792607036955070896285410, 9.703768129417084126986628733584, 11.50656761677525870193740618190, 11.88379976874163163997560281822, 13.55270254406507274149979798149, 14.25386858388551989938739815372