L(s) = 1 | + (−0.342 + 0.939i)2-s + (3.24 − 1.18i)3-s + (−0.766 − 0.642i)4-s + (0.984 + 0.173i)5-s + 3.45i·6-s + (−0.802 + 4.55i)7-s + (0.866 − 0.500i)8-s + (6.83 − 5.73i)9-s + (−0.5 + 0.866i)10-s + (−0.349 − 0.606i)11-s + (−3.24 − 1.18i)12-s + (−2.84 + 3.39i)13-s + (−4.00 − 2.31i)14-s + (3.40 − 0.599i)15-s + (0.173 + 0.984i)16-s + (−1.72 − 2.06i)17-s + ⋯ |
L(s) = 1 | + (−0.241 + 0.664i)2-s + (1.87 − 0.681i)3-s + (−0.383 − 0.321i)4-s + (0.440 + 0.0776i)5-s + 1.40i·6-s + (−0.303 + 1.72i)7-s + (0.306 − 0.176i)8-s + (2.27 − 1.91i)9-s + (−0.158 + 0.273i)10-s + (−0.105 − 0.182i)11-s + (−0.936 − 0.340i)12-s + (−0.789 + 0.940i)13-s + (−1.06 − 0.617i)14-s + (0.877 − 0.154i)15-s + (0.0434 + 0.246i)16-s + (−0.419 − 0.499i)17-s + ⋯ |
Λ(s)=(=(370s/2ΓC(s)L(s)(0.850−0.526i)Λ(2−s)
Λ(s)=(=(370s/2ΓC(s+1/2)L(s)(0.850−0.526i)Λ(1−s)
Degree: |
2 |
Conductor: |
370
= 2⋅5⋅37
|
Sign: |
0.850−0.526i
|
Analytic conductor: |
2.95446 |
Root analytic conductor: |
1.71885 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ370(21,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 370, ( :1/2), 0.850−0.526i)
|
Particular Values
L(1) |
≈ |
2.03131+0.577616i |
L(21) |
≈ |
2.03131+0.577616i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.342−0.939i)T |
| 5 | 1+(−0.984−0.173i)T |
| 37 | 1+(5.33+2.92i)T |
good | 3 | 1+(−3.24+1.18i)T+(2.29−1.92i)T2 |
| 7 | 1+(0.802−4.55i)T+(−6.57−2.39i)T2 |
| 11 | 1+(0.349+0.606i)T+(−5.5+9.52i)T2 |
| 13 | 1+(2.84−3.39i)T+(−2.25−12.8i)T2 |
| 17 | 1+(1.72+2.06i)T+(−2.95+16.7i)T2 |
| 19 | 1+(0.965+2.65i)T+(−14.5+12.2i)T2 |
| 23 | 1+(3.45+1.99i)T+(11.5+19.9i)T2 |
| 29 | 1+(−3.19+1.84i)T+(14.5−25.1i)T2 |
| 31 | 1+6.25iT−31T2 |
| 41 | 1+(−6.98−5.86i)T+(7.11+40.3i)T2 |
| 43 | 1+2.50iT−43T2 |
| 47 | 1+(4.65−8.05i)T+(−23.5−40.7i)T2 |
| 53 | 1+(−0.0217−0.123i)T+(−49.8+18.1i)T2 |
| 59 | 1+(−0.819+0.144i)T+(55.4−20.1i)T2 |
| 61 | 1+(1.85−2.20i)T+(−10.5−60.0i)T2 |
| 67 | 1+(1.45−8.27i)T+(−62.9−22.9i)T2 |
| 71 | 1+(−3.08+1.12i)T+(54.3−45.6i)T2 |
| 73 | 1+4.53T+73T2 |
| 79 | 1+(9.31+1.64i)T+(74.2+27.0i)T2 |
| 83 | 1+(1.52−1.28i)T+(14.4−81.7i)T2 |
| 89 | 1+(−0.886+0.156i)T+(83.6−30.4i)T2 |
| 97 | 1+(−13.1−7.61i)T+(48.5+84.0i)T2 |
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
−11.79451595070348736714489423214, −9.857831175772308066729309808234, −9.249963180140357054905923239543, −8.780444737989553009981229924985, −7.891807788396148645143166610405, −6.87350396915702463082999835207, −6.07385658132718518287367984621, −4.46519084596470775765474102312, −2.75476159765869590829832291840, −2.11299691117919043779833386591,
1.74747218039184994068121122479, 3.09015515176339869177734492422, 3.88569539011535471115523178879, 4.82748733347837818512588744051, 7.07680350642931821846229126914, 7.85260420721790030485476448174, 8.651129812271473952810117081977, 9.732619948000887391121720736911, 10.27501852600806947582607429811, 10.60500375340435595713653075962