L(s) = 1 | + (1.13 − 0.841i)2-s + (−1.86 − 1.86i)3-s + (0.582 − 1.91i)4-s + (−1.17 + 1.90i)5-s + (−3.68 − 0.547i)6-s + 3.61·7-s + (−0.949 − 2.66i)8-s + 3.92i·9-s + (0.271 + 3.15i)10-s + (−0.0947 − 0.0947i)11-s + (−4.64 + 2.47i)12-s + (2.59 + 2.59i)13-s + (4.10 − 3.04i)14-s + (5.72 − 1.36i)15-s + (−3.32 − 2.22i)16-s + 1.89i·17-s + ⋯ |
L(s) = 1 | + (0.803 − 0.595i)2-s + (−1.07 − 1.07i)3-s + (0.291 − 0.956i)4-s + (−0.524 + 0.851i)5-s + (−1.50 − 0.223i)6-s + 1.36·7-s + (−0.335 − 0.941i)8-s + 1.30i·9-s + (0.0858 + 0.996i)10-s + (−0.0285 − 0.0285i)11-s + (−1.34 + 0.714i)12-s + (0.719 + 0.719i)13-s + (1.09 − 0.813i)14-s + (1.47 − 0.351i)15-s + (−0.830 − 0.556i)16-s + 0.460i·17-s + ⋯ |
Λ(s)=(=(80s/2ΓC(s)L(s)(0.0356+0.999i)Λ(2−s)
Λ(s)=(=(80s/2ΓC(s+1/2)L(s)(0.0356+0.999i)Λ(1−s)
Degree: |
2 |
Conductor: |
80
= 24⋅5
|
Sign: |
0.0356+0.999i
|
Analytic conductor: |
0.638803 |
Root analytic conductor: |
0.799251 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ80(69,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 80, ( :1/2), 0.0356+0.999i)
|
Particular Values
L(1) |
≈ |
0.772669−0.745583i |
L(21) |
≈ |
0.772669−0.745583i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−1.13+0.841i)T |
| 5 | 1+(1.17−1.90i)T |
good | 3 | 1+(1.86+1.86i)T+3iT2 |
| 7 | 1−3.61T+7T2 |
| 11 | 1+(0.0947+0.0947i)T+11iT2 |
| 13 | 1+(−2.59−2.59i)T+13iT2 |
| 17 | 1−1.89iT−17T2 |
| 19 | 1+(2.16−2.16i)T−19iT2 |
| 23 | 1+5.08T+23T2 |
| 29 | 1+(−1.25+1.25i)T−29iT2 |
| 31 | 1+1.27T+31T2 |
| 37 | 1+(−2.25+2.25i)T−37iT2 |
| 41 | 1−8.52iT−41T2 |
| 43 | 1+(−1.61+1.61i)T−43iT2 |
| 47 | 1+2.53iT−47T2 |
| 53 | 1+(−5.67+5.67i)T−53iT2 |
| 59 | 1+(7.81+7.81i)T+59iT2 |
| 61 | 1+(−3.46+3.46i)T−61iT2 |
| 67 | 1+(−6.29−6.29i)T+67iT2 |
| 71 | 1−11.3iT−71T2 |
| 73 | 1+16.1T+73T2 |
| 79 | 1−1.13T+79T2 |
| 83 | 1+(3.75+3.75i)T+83iT2 |
| 89 | 1+3.98iT−89T2 |
| 97 | 1+10.3iT−97T2 |
show more | |
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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
−14.05794134165821605915364406931, −12.85581539311948702786602787864, −11.67506511313990209151599122266, −11.41864009601521384996493047447, −10.43270184290162603071521968431, −8.053540303558593094235578730035, −6.75419671620032411042743437130, −5.79531599990083030304929734926, −4.20043566813844357610118406853, −1.80668676801464873908425654601,
4.10604551183961593520974466299, 4.91890419466377989052123064061, 5.82108652583974171396891983396, 7.73951864899361091134915922247, 8.815593561223988286420934456126, 10.70570499765061518694295568104, 11.51668240227791420789817771715, 12.31609816488414275459646529925, 13.68470753231209521201716079442, 15.01126885590924162123683135856