L(s) = 1 | + (−0.923 + 0.382i)2-s + (−2.12 − 1.42i)3-s + (0.707 − 0.707i)4-s + (2.51 + 0.499i)6-s + (−0.0411 − 0.00818i)7-s + (−0.382 + 0.923i)8-s + (1.36 + 3.29i)9-s + (0.476 − 2.39i)11-s + (−2.51 + 0.499i)12-s + 2.17·13-s + (0.0411 − 0.00818i)14-s − i·16-s + (2.37 − 3.36i)17-s + (−2.51 − 2.51i)18-s + (−1.43 + 3.45i)19-s + ⋯ |
L(s) = 1 | + (−0.653 + 0.270i)2-s + (−1.22 − 0.821i)3-s + (0.353 − 0.353i)4-s + (1.02 + 0.204i)6-s + (−0.0155 − 0.00309i)7-s + (−0.135 + 0.326i)8-s + (0.454 + 1.09i)9-s + (0.143 − 0.721i)11-s + (−0.725 + 0.144i)12-s + 0.601·13-s + (0.0109 − 0.00218i)14-s − 0.250i·16-s + (0.576 − 0.816i)17-s + (−0.593 − 0.593i)18-s + (−0.328 + 0.792i)19-s + ⋯ |
Λ(s)=(=(850s/2ΓC(s)L(s)(−0.245+0.969i)Λ(2−s)
Λ(s)=(=(850s/2ΓC(s+1/2)L(s)(−0.245+0.969i)Λ(1−s)
Degree: |
2 |
Conductor: |
850
= 2⋅52⋅17
|
Sign: |
−0.245+0.969i
|
Analytic conductor: |
6.78728 |
Root analytic conductor: |
2.60524 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ850(193,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 850, ( :1/2), −0.245+0.969i)
|
Particular Values
L(1) |
≈ |
0.374478−0.480961i |
L(21) |
≈ |
0.374478−0.480961i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.923−0.382i)T |
| 5 | 1 |
| 17 | 1+(−2.37+3.36i)T |
good | 3 | 1+(2.12+1.42i)T+(1.14+2.77i)T2 |
| 7 | 1+(0.0411+0.00818i)T+(6.46+2.67i)T2 |
| 11 | 1+(−0.476+2.39i)T+(−10.1−4.20i)T2 |
| 13 | 1−2.17T+13T2 |
| 19 | 1+(1.43−3.45i)T+(−13.4−13.4i)T2 |
| 23 | 1+(−3.96−5.93i)T+(−8.80+21.2i)T2 |
| 29 | 1+(−4.73+7.08i)T+(−11.0−26.7i)T2 |
| 31 | 1+(−0.226−1.13i)T+(−28.6+11.8i)T2 |
| 37 | 1+(−4.07+6.10i)T+(−14.1−34.1i)T2 |
| 41 | 1+(−0.400−0.599i)T+(−15.6+37.8i)T2 |
| 43 | 1+(8.37+3.46i)T+(30.4+30.4i)T2 |
| 47 | 1+4.13iT−47T2 |
| 53 | 1+(2.01+4.87i)T+(−37.4+37.4i)T2 |
| 59 | 1+(8.77−3.63i)T+(41.7−41.7i)T2 |
| 61 | 1+(−5.87+3.92i)T+(23.3−56.3i)T2 |
| 67 | 1+(11.0+11.0i)T+67iT2 |
| 71 | 1+(−5.22+1.04i)T+(65.5−27.1i)T2 |
| 73 | 1+(11.1−2.22i)T+(67.4−27.9i)T2 |
| 79 | 1+(7.04+1.40i)T+(72.9+30.2i)T2 |
| 83 | 1+(−15.6+6.46i)T+(58.6−58.6i)T2 |
| 89 | 1+(−11.5+11.5i)T−89iT2 |
| 97 | 1+(−4.02+0.800i)T+(89.6−37.1i)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
−10.02665806406109035978324007402, −9.061107637480024257334979133218, −8.054603876458207170910102201224, −7.34916618509026901483985859975, −6.39780977965483882917652449827, −5.88491664900162869685091017738, −4.99967971810615021778250274597, −3.35041286374311290621614964777, −1.62942479661605859662428200100, −0.51408245678426111726077653120,
1.21569721416978961078343875859, 2.97470488577860631386252467867, 4.31439596250327602947984064266, 4.99014714000913374154211746020, 6.22571634549238424093090007297, 6.77083217780988735493591888149, 8.063829174162432076913170229242, 8.944760870258222702070941667944, 9.812836338240012279957954625128, 10.55911977571246952257652691479