L(s) = 1 | + (4.59 + 7.95i)2-s + (−4.5 + 7.79i)3-s + (−26.1 + 45.2i)4-s + (−11.0 − 19.1i)5-s − 82.6·6-s + (126. + 26.6i)7-s − 186.·8-s + (−40.5 − 70.1i)9-s + (101. − 175. i)10-s + (−208. + 360. i)11-s + (−235. − 407. i)12-s + 797.·13-s + (370. + 1.13e3i)14-s + 198.·15-s + (−18.6 − 32.3i)16-s + (687. − 1.19e3i)17-s + ⋯ |
L(s) = 1 | + (0.811 + 1.40i)2-s + (−0.288 + 0.499i)3-s + (−0.817 + 1.41i)4-s + (−0.197 − 0.341i)5-s − 0.937·6-s + (0.978 + 0.205i)7-s − 1.02·8-s + (−0.166 − 0.288i)9-s + (0.320 − 0.554i)10-s + (−0.519 + 0.899i)11-s + (−0.471 − 0.817i)12-s + 1.30·13-s + (0.505 + 1.54i)14-s + 0.227·15-s + (−0.0182 − 0.0315i)16-s + (0.577 − 0.999i)17-s + ⋯ |
Λ(s)=(=(21s/2ΓC(s)L(s)(−0.701−0.712i)Λ(6−s)
Λ(s)=(=(21s/2ΓC(s+5/2)L(s)(−0.701−0.712i)Λ(1−s)
Degree: |
2 |
Conductor: |
21
= 3⋅7
|
Sign: |
−0.701−0.712i
|
Analytic conductor: |
3.36806 |
Root analytic conductor: |
1.83522 |
Motivic weight: |
5 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ21(4,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 21, ( :5/2), −0.701−0.712i)
|
Particular Values
L(3) |
≈ |
0.739914+1.76562i |
L(21) |
≈ |
0.739914+1.76562i |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(4.5−7.79i)T |
| 7 | 1+(−126.−26.6i)T |
good | 2 | 1+(−4.59−7.95i)T+(−16+27.7i)T2 |
| 5 | 1+(11.0+19.1i)T+(−1.56e3+2.70e3i)T2 |
| 11 | 1+(208.−360.i)T+(−8.05e4−1.39e5i)T2 |
| 13 | 1−797.T+3.71e5T2 |
| 17 | 1+(−687.+1.19e3i)T+(−7.09e5−1.22e6i)T2 |
| 19 | 1+(1.15e3+2.00e3i)T+(−1.23e6+2.14e6i)T2 |
| 23 | 1+(−477.−827.i)T+(−3.21e6+5.57e6i)T2 |
| 29 | 1+7.03e3T+2.05e7T2 |
| 31 | 1+(630.−1.09e3i)T+(−1.43e7−2.47e7i)T2 |
| 37 | 1+(4.88e3+8.46e3i)T+(−3.46e7+6.00e7i)T2 |
| 41 | 1+5.40e3T+1.15e8T2 |
| 43 | 1−1.96e4T+1.47e8T2 |
| 47 | 1+(1.02e3+1.78e3i)T+(−1.14e8+1.98e8i)T2 |
| 53 | 1+(9.01e3−1.56e4i)T+(−2.09e8−3.62e8i)T2 |
| 59 | 1+(3.71e3−6.43e3i)T+(−3.57e8−6.19e8i)T2 |
| 61 | 1+(1.74e3+3.02e3i)T+(−4.22e8+7.31e8i)T2 |
| 67 | 1+(7.92e3−1.37e4i)T+(−6.75e8−1.16e9i)T2 |
| 71 | 1−5.81e4T+1.80e9T2 |
| 73 | 1+(1.95e4−3.38e4i)T+(−1.03e9−1.79e9i)T2 |
| 79 | 1+(4.88e3+8.45e3i)T+(−1.53e9+2.66e9i)T2 |
| 83 | 1+7.03e4T+3.93e9T2 |
| 89 | 1+(7.21e4+1.24e5i)T+(−2.79e9+4.83e9i)T2 |
| 97 | 1+7.93e4T+8.58e9T2 |
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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
−17.19439882985066342462652274103, −15.95585913075066343180120649581, −15.26540180598407412080246546056, −14.08777643776746820189710207587, −12.72163279399230164755742844272, −11.05765279897405870925508428272, −8.792674763018713267488283483167, −7.33858803218183208350367612115, −5.50981562690680467665656021301, −4.40540228070355111938452123360,
1.50596623695673060604327933545, 3.68650261413003741380346724003, 5.68612600372883597512352142615, 8.148839507591337179809388701754, 10.69063636377348519966075350535, 11.19595284008474540462917354504, 12.59742753673654916291021060778, 13.66351218527995835355671435007, 14.77845794790042905049315471082, 16.81520701392112863019080952298