L(s) = 1 | + (0.382 + 0.923i)2-s + (−0.442 − 2.22i)3-s + (−0.707 + 0.707i)4-s + (1.88 − 1.26i)6-s + (1.93 − 1.29i)7-s + (−0.923 − 0.382i)8-s + (−1.98 + 0.820i)9-s + (1.86 + 2.78i)11-s + (1.88 + 1.26i)12-s + 4.73·13-s + (1.93 + 1.29i)14-s − i·16-s + (−3.36 − 2.38i)17-s + (−1.51 − 1.51i)18-s + (0.786 + 0.325i)19-s + ⋯ |
L(s) = 1 | + (0.270 + 0.653i)2-s + (−0.255 − 1.28i)3-s + (−0.353 + 0.353i)4-s + (0.769 − 0.514i)6-s + (0.731 − 0.488i)7-s + (−0.326 − 0.135i)8-s + (−0.660 + 0.273i)9-s + (0.561 + 0.840i)11-s + (0.544 + 0.363i)12-s + 1.31·13-s + (0.517 + 0.345i)14-s − 0.250i·16-s + (−0.815 − 0.578i)17-s + (−0.357 − 0.357i)18-s + (0.180 + 0.0747i)19-s + ⋯ |
Λ(s)=(=(850s/2ΓC(s)L(s)(0.767+0.641i)Λ(2−s)
Λ(s)=(=(850s/2ΓC(s+1/2)L(s)(0.767+0.641i)Λ(1−s)
Degree: |
2 |
Conductor: |
850
= 2⋅52⋅17
|
Sign: |
0.767+0.641i
|
Analytic conductor: |
6.78728 |
Root analytic conductor: |
2.60524 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ850(843,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 850, ( :1/2), 0.767+0.641i)
|
Particular Values
L(1) |
≈ |
1.67824−0.608860i |
L(21) |
≈ |
1.67824−0.608860i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.382−0.923i)T |
| 5 | 1 |
| 17 | 1+(3.36+2.38i)T |
good | 3 | 1+(0.442+2.22i)T+(−2.77+1.14i)T2 |
| 7 | 1+(−1.93+1.29i)T+(2.67−6.46i)T2 |
| 11 | 1+(−1.86−2.78i)T+(−4.20+10.1i)T2 |
| 13 | 1−4.73T+13T2 |
| 19 | 1+(−0.786−0.325i)T+(13.4+13.4i)T2 |
| 23 | 1+(4.12+0.820i)T+(21.2+8.80i)T2 |
| 29 | 1+(−3.83+0.762i)T+(26.7−11.0i)T2 |
| 31 | 1+(−4.45+6.66i)T+(−11.8−28.6i)T2 |
| 37 | 1+(−9.93+1.97i)T+(34.1−14.1i)T2 |
| 41 | 1+(0.472+0.0940i)T+(37.8+15.6i)T2 |
| 43 | 1+(0.702−1.69i)T+(−30.4−30.4i)T2 |
| 47 | 1+6.52iT−47T2 |
| 53 | 1+(3.22−1.33i)T+(37.4−37.4i)T2 |
| 59 | 1+(5.52+13.3i)T+(−41.7+41.7i)T2 |
| 61 | 1+(1.17−5.90i)T+(−56.3−23.3i)T2 |
| 67 | 1+(−0.912−0.912i)T+67iT2 |
| 71 | 1+(−10.9−7.33i)T+(27.1+65.5i)T2 |
| 73 | 1+(10.1+6.77i)T+(27.9+67.4i)T2 |
| 79 | 1+(−6.77+4.52i)T+(30.2−72.9i)T2 |
| 83 | 1+(3.72+8.98i)T+(−58.6+58.6i)T2 |
| 89 | 1+(2.75−2.75i)T−89iT2 |
| 97 | 1+(−7.64−5.10i)T+(37.1+89.6i)T2 |
show more | |
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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
−10.03197019496169168831845041946, −8.978908719075742508574841460883, −7.954566712931030284724226822414, −7.57548082999713028868038222777, −6.46903305393840917647153188350, −6.20806897414937869580701338355, −4.76045124175921427944932153392, −3.97416595797263946294554743314, −2.20099098000632130502861244690, −0.977764236950555286324649862660,
1.43670839602344374373775648483, 3.04267092922784627763575169652, 4.02353850746911936133837885265, 4.65633228124694624887690418604, 5.70414260795856169417207432257, 6.39657759614317088863934157769, 8.234218807928529247506070614454, 8.735599649689829582329450107801, 9.580161632461479076186701689807, 10.47886307710487813061876860880