L(s) = 1 | + (0.176 + 0.176i)2-s + (−1.51 − 0.629i)3-s − 1.93i·4-s + (1.62 − 1.53i)5-s + (−0.156 − 0.377i)6-s + (0.547 + 1.32i)7-s + (0.693 − 0.693i)8-s + (−0.210 − 0.210i)9-s + (0.556 + 0.0165i)10-s + (1.03 + 2.48i)11-s + (−1.21 + 2.94i)12-s + 0.174·13-s + (−0.136 + 0.329i)14-s + (−3.43 + 1.30i)15-s − 3.63·16-s + (2.27 + 3.44i)17-s + ⋯ |
L(s) = 1 | + (0.124 + 0.124i)2-s + (−0.876 − 0.363i)3-s − 0.969i·4-s + (0.727 − 0.685i)5-s + (−0.0639 − 0.154i)6-s + (0.206 + 0.499i)7-s + (0.245 − 0.245i)8-s + (−0.0703 − 0.0703i)9-s + (0.175 + 0.00523i)10-s + (0.310 + 0.750i)11-s + (−0.351 + 0.849i)12-s + 0.0484·13-s + (−0.0364 + 0.0879i)14-s + (−0.887 + 0.336i)15-s − 0.908·16-s + (0.550 + 0.834i)17-s + ⋯ |
Λ(s)=(=(85s/2ΓC(s)L(s)(0.588+0.808i)Λ(2−s)
Λ(s)=(=(85s/2ΓC(s+1/2)L(s)(0.588+0.808i)Λ(1−s)
Degree: |
2 |
Conductor: |
85
= 5⋅17
|
Sign: |
0.588+0.808i
|
Analytic conductor: |
0.678728 |
Root analytic conductor: |
0.823849 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ85(59,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 85, ( :1/2), 0.588+0.808i)
|
Particular Values
L(1) |
≈ |
0.790864−0.402276i |
L(21) |
≈ |
0.790864−0.402276i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1+(−1.62+1.53i)T |
| 17 | 1+(−2.27−3.44i)T |
good | 2 | 1+(−0.176−0.176i)T+2iT2 |
| 3 | 1+(1.51+0.629i)T+(2.12+2.12i)T2 |
| 7 | 1+(−0.547−1.32i)T+(−4.94+4.94i)T2 |
| 11 | 1+(−1.03−2.48i)T+(−7.77+7.77i)T2 |
| 13 | 1−0.174T+13T2 |
| 19 | 1+(2.69−2.69i)T−19iT2 |
| 23 | 1+(−2.54+1.05i)T+(16.2−16.2i)T2 |
| 29 | 1+(−5.94−2.46i)T+(20.5+20.5i)T2 |
| 31 | 1+(0.188−0.454i)T+(−21.9−21.9i)T2 |
| 37 | 1+(5.30+2.19i)T+(26.1+26.1i)T2 |
| 41 | 1+(−5.54+2.29i)T+(28.9−28.9i)T2 |
| 43 | 1+(8.00−8.00i)T−43iT2 |
| 47 | 1−2.65T+47T2 |
| 53 | 1+(8.73+8.73i)T+53iT2 |
| 59 | 1+(5.72+5.72i)T+59iT2 |
| 61 | 1+(6.41−2.65i)T+(43.1−43.1i)T2 |
| 67 | 1+12.3iT−67T2 |
| 71 | 1+(4.67−11.2i)T+(−50.2−50.2i)T2 |
| 73 | 1+(5.99−14.4i)T+(−51.6−51.6i)T2 |
| 79 | 1+(4.94+11.9i)T+(−55.8+55.8i)T2 |
| 83 | 1+(−6.06−6.06i)T+83iT2 |
| 89 | 1−11.4iT−89T2 |
| 97 | 1+(0.879−2.12i)T+(−68.5−68.5i)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
−14.22878229174189234290650152262, −12.81305143459083596544905245985, −12.14115272550050885366180279703, −10.80370975621283226437015863879, −9.794941517071743685337967770981, −8.630278816642640461115755766497, −6.62103284979673782865944163306, −5.80528994964535584891758735969, −4.83020201582188431581469610648, −1.55870074942881671037913755154,
2.99467006010778839180334156545, 4.72000607096478589500387763141, 6.16469478457576260365743193729, 7.40084440872480146159075586997, 8.885961996383748209899652919131, 10.40201842860853381902669063420, 11.16797588666866726449340424260, 12.03603254148827562701427630983, 13.51263622640938331255135260043, 14.05738940719631618012341036375