L(s) = 1 | + (0.395 + 0.228i)2-s − 2.79·3-s + (−0.895 − 1.55i)4-s + (0.395 − 0.228i)5-s + (−1.10 − 0.637i)6-s − 1.73i·8-s + 4.79·9-s + 0.208·10-s − 3.92i·11-s + (2.5 + 4.33i)12-s + (−3.5 − 0.866i)13-s + (−1.10 + 0.637i)15-s + (−1.39 + 2.41i)16-s + (1.5 + 2.59i)17-s + (1.89 + 1.09i)18-s + 1.37i·19-s + ⋯ |
L(s) = 1 | + (0.279 + 0.161i)2-s − 1.61·3-s + (−0.447 − 0.775i)4-s + (0.176 − 0.102i)5-s + (−0.450 − 0.260i)6-s − 0.612i·8-s + 1.59·9-s + 0.0660·10-s − 1.18i·11-s + (0.721 + 1.24i)12-s + (−0.970 − 0.240i)13-s + (−0.285 + 0.164i)15-s + (−0.348 + 0.604i)16-s + (0.363 + 0.630i)17-s + (0.446 + 0.257i)18-s + 0.314i·19-s + ⋯ |
Λ(s)=(=(637s/2ΓC(s)L(s)(−0.794−0.606i)Λ(2−s)
Λ(s)=(=(637s/2ΓC(s+1/2)L(s)(−0.794−0.606i)Λ(1−s)
Degree: |
2 |
Conductor: |
637
= 72⋅13
|
Sign: |
−0.794−0.606i
|
Analytic conductor: |
5.08647 |
Root analytic conductor: |
2.25532 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ637(361,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
1
|
Selberg data: |
(2, 637, ( :1/2), −0.794−0.606i)
|
Particular Values
L(1) |
= |
0 |
L(21) |
= |
0 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 7 | 1 |
| 13 | 1+(3.5+0.866i)T |
good | 2 | 1+(−0.395−0.228i)T+(1+1.73i)T2 |
| 3 | 1+2.79T+3T2 |
| 5 | 1+(−0.395+0.228i)T+(2.5−4.33i)T2 |
| 11 | 1+3.92iT−11T2 |
| 17 | 1+(−1.5−2.59i)T+(−8.5+14.7i)T2 |
| 19 | 1−1.37iT−19T2 |
| 23 | 1+(−0.791+1.37i)T+(−11.5−19.9i)T2 |
| 29 | 1+(−3.39−5.88i)T+(−14.5+25.1i)T2 |
| 31 | 1+(7.5+4.33i)T+(15.5+26.8i)T2 |
| 37 | 1+(6+3.46i)T+(18.5+32.0i)T2 |
| 41 | 1+(6.79−3.92i)T+(20.5−35.5i)T2 |
| 43 | 1+(4.68−8.11i)T+(−21.5−37.2i)T2 |
| 47 | 1+(8.29−4.78i)T+(23.5−40.7i)T2 |
| 53 | 1+(3.08−5.33i)T+(−26.5−45.8i)T2 |
| 59 | 1+(−10.6+6.15i)T+(29.5−51.0i)T2 |
| 61 | 1−14.7T+61T2 |
| 67 | 1+4.47iT−67T2 |
| 71 | 1+(−3.79−2.18i)T+(35.5+61.4i)T2 |
| 73 | 1+(−3−1.73i)T+(36.5+63.2i)T2 |
| 79 | 1+(−3−5.19i)T+(−39.5+68.4i)T2 |
| 83 | 1+7.02iT−83T2 |
| 89 | 1+(13.9+8.07i)T+(44.5+77.0i)T2 |
| 97 | 1+(−6.31−3.64i)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
−10.20743402819269398667772228648, −9.514966218346523668108605171196, −8.303402142257381408112723300160, −6.95645538376390165431511026074, −6.17124881041506771497473506334, −5.40875844360904347246112804037, −4.98262095179010938913852051827, −3.63688282479232588270682973216, −1.38228298275911195473528000535, 0,
2.19183216262703043005804863449, 3.83316924665352311824692119410, 5.00027443910550228092071050657, 5.21519282429262463344276603999, 6.80135741074868912472326485978, 7.18905558440256622046945218427, 8.456586967780593685694132406645, 9.768161421122547893614309885470, 10.18236098644826566047376551066