L(s) = 1 | + (−0.309 + 0.951i)2-s + (−0.126 − 0.0917i)3-s + (−0.809 − 0.587i)4-s + (−1.23 + 1.86i)5-s + (0.126 − 0.0917i)6-s − 2.41·7-s + (0.809 − 0.587i)8-s + (−0.919 − 2.83i)9-s + (−1.39 − 1.74i)10-s + (−0.733 + 2.25i)11-s + (0.0482 + 0.148i)12-s + (−0.309 − 0.951i)13-s + (0.747 − 2.30i)14-s + (0.326 − 0.122i)15-s + (0.309 + 0.951i)16-s + (4.87 − 3.54i)17-s + ⋯ |
L(s) = 1 | + (−0.218 + 0.672i)2-s + (−0.0728 − 0.0529i)3-s + (−0.404 − 0.293i)4-s + (−0.551 + 0.833i)5-s + (0.0515 − 0.0374i)6-s − 0.914·7-s + (0.286 − 0.207i)8-s + (−0.306 − 0.943i)9-s + (−0.440 − 0.553i)10-s + (−0.221 + 0.680i)11-s + (0.0139 + 0.0428i)12-s + (−0.0857 − 0.263i)13-s + (0.199 − 0.614i)14-s + (0.0843 − 0.0315i)15-s + (0.0772 + 0.237i)16-s + (1.18 − 0.858i)17-s + ⋯ |
Λ(s)=(=(650s/2ΓC(s)L(s)(0.607+0.794i)Λ(2−s)
Λ(s)=(=(650s/2ΓC(s+1/2)L(s)(0.607+0.794i)Λ(1−s)
Degree: |
2 |
Conductor: |
650
= 2⋅52⋅13
|
Sign: |
0.607+0.794i
|
Analytic conductor: |
5.19027 |
Root analytic conductor: |
2.27821 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ650(521,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 650, ( :1/2), 0.607+0.794i)
|
Particular Values
L(1) |
≈ |
0.524911−0.259439i |
L(21) |
≈ |
0.524911−0.259439i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.309−0.951i)T |
| 5 | 1+(1.23−1.86i)T |
| 13 | 1+(0.309+0.951i)T |
good | 3 | 1+(0.126+0.0917i)T+(0.927+2.85i)T2 |
| 7 | 1+2.41T+7T2 |
| 11 | 1+(0.733−2.25i)T+(−8.89−6.46i)T2 |
| 17 | 1+(−4.87+3.54i)T+(5.25−16.1i)T2 |
| 19 | 1+(−1.61+1.17i)T+(5.87−18.0i)T2 |
| 23 | 1+(−2.62+8.08i)T+(−18.6−13.5i)T2 |
| 29 | 1+(−0.670−0.486i)T+(8.96+27.5i)T2 |
| 31 | 1+(7.09−5.15i)T+(9.57−29.4i)T2 |
| 37 | 1+(1.19+3.66i)T+(−29.9+21.7i)T2 |
| 41 | 1+(2.20+6.78i)T+(−33.1+24.0i)T2 |
| 43 | 1−12.5T+43T2 |
| 47 | 1+(4.58+3.32i)T+(14.5+44.6i)T2 |
| 53 | 1+(5.91+4.29i)T+(16.3+50.4i)T2 |
| 59 | 1+(−0.383−1.17i)T+(−47.7+34.6i)T2 |
| 61 | 1+(−1.27+3.91i)T+(−49.3−35.8i)T2 |
| 67 | 1+(2.06−1.49i)T+(20.7−63.7i)T2 |
| 71 | 1+(5.50+4.00i)T+(21.9+67.5i)T2 |
| 73 | 1+(0.613−1.88i)T+(−59.0−42.9i)T2 |
| 79 | 1+(−7.13−5.18i)T+(24.4+75.1i)T2 |
| 83 | 1+(−6.58+4.78i)T+(25.6−78.9i)T2 |
| 89 | 1+(−0.650+2.00i)T+(−72.0−52.3i)T2 |
| 97 | 1+(10.7+7.77i)T+(29.9+92.2i)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
−10.26117637878456632502344539752, −9.533958669318932747459062446546, −8.702106821830962555954800910341, −7.46507188014734972757837760917, −6.99086989819347481350419937743, −6.18233667125640657547976365631, −5.09597023776910059268099252596, −3.70453804185645627281125363715, −2.84160598927255381882949858309, −0.36792676686944837507364858791,
1.38107114072406274861837328251, 3.06492051627789575366017243539, 3.88604686805847829570970358424, 5.17056738922424776207964746051, 5.92067566326939009496847998892, 7.60549602290586884695644806787, 8.032526610606714870170725592701, 9.158942638859230275144792197195, 9.725936095209995023132323693832, 10.78669637673034567890850071216