L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.866 + 1.5i)3-s + (−0.500 + 0.866i)4-s + 1.73i·6-s + (−2.59 + 1.5i)7-s + 3i·8-s + (−1.5 − 2.59i)9-s + (1 + 1.73i)11-s + (−0.866 − 1.5i)12-s + (−1.73 − i)13-s + (−1.5 + 2.59i)14-s + (0.500 + 0.866i)16-s + 4i·17-s + (−2.59 − 1.5i)18-s + 8·19-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.499 + 0.866i)3-s + (−0.250 + 0.433i)4-s + 0.707i·6-s + (−0.981 + 0.566i)7-s + 1.06i·8-s + (−0.5 − 0.866i)9-s + (0.301 + 0.522i)11-s + (−0.249 − 0.433i)12-s + (−0.480 − 0.277i)13-s + (−0.400 + 0.694i)14-s + (0.125 + 0.216i)16-s + 0.970i·17-s + (−0.612 − 0.353i)18-s + 1.83·19-s + ⋯ |
Λ(s)=(=(225s/2ΓC(s)L(s)(−0.232−0.972i)Λ(2−s)
Λ(s)=(=(225s/2ΓC(s+1/2)L(s)(−0.232−0.972i)Λ(1−s)
Degree: |
2 |
Conductor: |
225
= 32⋅52
|
Sign: |
−0.232−0.972i
|
Analytic conductor: |
1.79663 |
Root analytic conductor: |
1.34038 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ225(49,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 225, ( :1/2), −0.232−0.972i)
|
Particular Values
L(1) |
≈ |
0.663235+0.840329i |
L(21) |
≈ |
0.663235+0.840329i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(0.866−1.5i)T |
| 5 | 1 |
good | 2 | 1+(−0.866+0.5i)T+(1−1.73i)T2 |
| 7 | 1+(2.59−1.5i)T+(3.5−6.06i)T2 |
| 11 | 1+(−1−1.73i)T+(−5.5+9.52i)T2 |
| 13 | 1+(1.73+i)T+(6.5+11.2i)T2 |
| 17 | 1−4iT−17T2 |
| 19 | 1−8T+19T2 |
| 23 | 1+(−2.59−1.5i)T+(11.5+19.9i)T2 |
| 29 | 1+(0.5+0.866i)T+(−14.5+25.1i)T2 |
| 31 | 1+(−15.5−26.8i)T2 |
| 37 | 1+4iT−37T2 |
| 41 | 1+(2.5−4.33i)T+(−20.5−35.5i)T2 |
| 43 | 1+(−6.92+4i)T+(21.5−37.2i)T2 |
| 47 | 1+(−6.06+3.5i)T+(23.5−40.7i)T2 |
| 53 | 1−2iT−53T2 |
| 59 | 1+(7−12.1i)T+(−29.5−51.0i)T2 |
| 61 | 1+(3.5+6.06i)T+(−30.5+52.8i)T2 |
| 67 | 1+(−2.59−1.5i)T+(33.5+58.0i)T2 |
| 71 | 1−2T+71T2 |
| 73 | 1+4iT−73T2 |
| 79 | 1+(3+5.19i)T+(−39.5+68.4i)T2 |
| 83 | 1+(7.79−4.5i)T+(41.5−71.8i)T2 |
| 89 | 1−15T+89T2 |
| 97 | 1+(−1.73+i)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
−12.29760985805107917983992249886, −11.87981110438293629006701230438, −10.69425878423511460153382332001, −9.602821310844246444855257997651, −8.973711634705946568150305396738, −7.47395685017942830436648964286, −5.97373834221418650207816815852, −5.08743014639196802501939699801, −3.87077696061692530105290520323, −2.92472370356700460003154624998,
0.815525097298749108345657997160, 3.20299101371822093947110918609, 4.82443299653086056644195378616, 5.81671718120177359628184083453, 6.79013835701274226589800914758, 7.43938691140677565904046477327, 9.169397830076688743504235621598, 10.00654806075417124756598694700, 11.20175042978215047880288400662, 12.20249156234456946547407791098