L(s) = 1 | + (3.19 + 5.52i)2-s + (2.91 + 5.04i)3-s + (235. − 408. i)4-s − 1.06e3·5-s + (−18.5 + 32.2i)6-s + (3.23e3 − 5.60e3i)7-s + 6.27e3·8-s + (9.82e3 − 1.70e4i)9-s + (−3.39e3 − 5.88e3i)10-s + (−9.31e3 − 1.61e4i)11-s + 2.74e3·12-s + (7.63e4 − 6.90e4i)13-s + 4.13e4·14-s + (−3.09e3 − 5.36e3i)15-s + (−1.00e5 − 1.74e5i)16-s + (−2.60e5 + 4.52e5i)17-s + ⋯ |
L(s) = 1 | + (0.141 + 0.244i)2-s + (0.0207 + 0.0359i)3-s + (0.460 − 0.797i)4-s − 0.761·5-s + (−0.00585 + 0.0101i)6-s + (0.509 − 0.882i)7-s + 0.541·8-s + (0.499 − 0.864i)9-s + (−0.107 − 0.185i)10-s + (−0.191 − 0.332i)11-s + 0.0382·12-s + (0.741 − 0.670i)13-s + 0.287·14-s + (−0.0158 − 0.0273i)15-s + (−0.383 − 0.664i)16-s + (−0.757 + 1.31i)17-s + ⋯ |
Λ(s)=(=(13s/2ΓC(s)L(s)(0.461+0.886i)Λ(10−s)
Λ(s)=(=(13s/2ΓC(s+9/2)L(s)(0.461+0.886i)Λ(1−s)
Degree: |
2 |
Conductor: |
13
|
Sign: |
0.461+0.886i
|
Analytic conductor: |
6.69546 |
Root analytic conductor: |
2.58755 |
Motivic weight: |
9 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ13(3,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 13, ( :9/2), 0.461+0.886i)
|
Particular Values
L(5) |
≈ |
1.50181−0.911191i |
L(21) |
≈ |
1.50181−0.911191i |
L(211) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 13 | 1+(−7.63e4+6.90e4i)T |
good | 2 | 1+(−3.19−5.52i)T+(−256+443.i)T2 |
| 3 | 1+(−2.91−5.04i)T+(−9.84e3+1.70e4i)T2 |
| 5 | 1+1.06e3T+1.95e6T2 |
| 7 | 1+(−3.23e3+5.60e3i)T+(−2.01e7−3.49e7i)T2 |
| 11 | 1+(9.31e3+1.61e4i)T+(−1.17e9+2.04e9i)T2 |
| 17 | 1+(2.60e5−4.52e5i)T+(−5.92e10−1.02e11i)T2 |
| 19 | 1+(1.36e5−2.37e5i)T+(−1.61e11−2.79e11i)T2 |
| 23 | 1+(−4.42e5−7.67e5i)T+(−9.00e11+1.55e12i)T2 |
| 29 | 1+(1.89e6+3.27e6i)T+(−7.25e12+1.25e13i)T2 |
| 31 | 1−9.87e6T+2.64e13T2 |
| 37 | 1+(−5.10e6−8.84e6i)T+(−6.49e13+1.12e14i)T2 |
| 41 | 1+(−6.37e6−1.10e7i)T+(−1.63e14+2.83e14i)T2 |
| 43 | 1+(2.01e6−3.49e6i)T+(−2.51e14−4.35e14i)T2 |
| 47 | 1−1.84e6T+1.11e15T2 |
| 53 | 1−5.32e7T+3.29e15T2 |
| 59 | 1+(−8.32e6+1.44e7i)T+(−4.33e15−7.50e15i)T2 |
| 61 | 1+(−4.50e7+7.79e7i)T+(−5.84e15−1.01e16i)T2 |
| 67 | 1+(1.30e7+2.25e7i)T+(−1.36e16+2.35e16i)T2 |
| 71 | 1+(1.40e8−2.42e8i)T+(−2.29e16−3.97e16i)T2 |
| 73 | 1−1.22e8T+5.88e16T2 |
| 79 | 1+4.73e8T+1.19e17T2 |
| 83 | 1−7.61e8T+1.86e17T2 |
| 89 | 1+(6.13e7+1.06e8i)T+(−1.75e17+3.03e17i)T2 |
| 97 | 1+(1.54e8−2.67e8i)T+(−3.80e17−6.58e17i)T2 |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−17.38739583197867975202272254005, −15.71415516789454939450454471482, −14.99465133376969635252997285163, −13.38237422398847672088447366298, −11.43615967380529443095532286541, −10.25782641757604214318167310958, −7.961279999318975844122005836111, −6.30130408858647237033894471436, −4.09573368744352287510834433970, −1.02391604874757114450532487375,
2.34329966717100102972544402589, 4.49428699417159863637617750974, 7.20719384804755308420776295939, 8.590412995038826874338153360985, 11.07210209192433145988525756964, 12.02284421561984009502331059125, 13.47652568288825034729775560652, 15.48066865405650025967289282079, 16.30418223244241988854972383950, 18.04288638886111515186374722593