L(s) = 1 | + (2.63 − 2.63i)2-s + (−0.614 − 0.614i)3-s − 9.92i·4-s + (−4.07 + 2.89i)5-s − 3.24·6-s + (1.62 − 1.62i)7-s + (−15.6 − 15.6i)8-s − 8.24i·9-s + (−3.12 + 18.3i)10-s + 12.6·11-s + (−6.09 + 6.09i)12-s + (13.1 + 13.1i)13-s − 8.57i·14-s + (4.28 + 0.728i)15-s − 42.7·16-s + (2.73 − 2.73i)17-s + ⋯ |
L(s) = 1 | + (1.31 − 1.31i)2-s + (−0.204 − 0.204i)3-s − 2.48i·4-s + (−0.815 + 0.578i)5-s − 0.540·6-s + (0.232 − 0.232i)7-s + (−1.95 − 1.95i)8-s − 0.916i·9-s + (−0.312 + 1.83i)10-s + 1.15·11-s + (−0.507 + 0.507i)12-s + (1.01 + 1.01i)13-s − 0.612i·14-s + (0.285 + 0.0485i)15-s − 2.67·16-s + (0.161 − 0.161i)17-s + ⋯ |
Λ(s)=(=(115s/2ΓC(s)L(s)(−0.660+0.750i)Λ(3−s)
Λ(s)=(=(115s/2ΓC(s+1)L(s)(−0.660+0.750i)Λ(1−s)
Degree: |
2 |
Conductor: |
115
= 5⋅23
|
Sign: |
−0.660+0.750i
|
Analytic conductor: |
3.13352 |
Root analytic conductor: |
1.77017 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ115(93,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 115, ( :1), −0.660+0.750i)
|
Particular Values
L(23) |
≈ |
0.935630−2.07072i |
L(21) |
≈ |
0.935630−2.07072i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1+(4.07−2.89i)T |
| 23 | 1+(−3.39−3.39i)T |
good | 2 | 1+(−2.63+2.63i)T−4iT2 |
| 3 | 1+(0.614+0.614i)T+9iT2 |
| 7 | 1+(−1.62+1.62i)T−49iT2 |
| 11 | 1−12.6T+121T2 |
| 13 | 1+(−13.1−13.1i)T+169iT2 |
| 17 | 1+(−2.73+2.73i)T−289iT2 |
| 19 | 1−26.9iT−361T2 |
| 29 | 1−4.50iT−841T2 |
| 31 | 1+38.0T+961T2 |
| 37 | 1+(−16.1+16.1i)T−1.36e3iT2 |
| 41 | 1+73.7T+1.68e3T2 |
| 43 | 1+(8.28+8.28i)T+1.84e3iT2 |
| 47 | 1+(−30.7+30.7i)T−2.20e3iT2 |
| 53 | 1+(−52.1−52.1i)T+2.80e3iT2 |
| 59 | 1+34.1iT−3.48e3T2 |
| 61 | 1+2.91T+3.72e3T2 |
| 67 | 1+(55.9−55.9i)T−4.48e3iT2 |
| 71 | 1+3.68T+5.04e3T2 |
| 73 | 1+(−86.8−86.8i)T+5.32e3iT2 |
| 79 | 1−52.8iT−6.24e3T2 |
| 83 | 1+(101.+101.i)T+6.88e3iT2 |
| 89 | 1+98.9iT−7.92e3T2 |
| 97 | 1+(−1.90+1.90i)T−9.40e3iT2 |
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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
−12.66303600886717431431585855213, −11.77312393809989755654058212830, −11.43829596249417405644979377808, −10.31060122769297766312161350946, −9.002638633213285170231600480356, −6.91252271543637810130176426748, −5.92003365444513692086572405246, −4.06332363882323828098151315994, −3.57666603032619257820527654543, −1.42790639382048308223840975176,
3.53127612234937543705044078907, 4.66929713534406634514530868483, 5.56248833969923676020511241597, 6.90134614818380212752433795715, 8.030562343415461348409284948130, 8.810736922915282795567463667114, 11.08031133862296560197899941051, 11.94100008523965940755898240508, 13.03388583710523791352985044664, 13.66622142043064869903338786669