L(s) = 1 | + (−0.707 + 0.707i)2-s + (1.46 − 0.920i)3-s − 1.00i·4-s + (−3.62 − 0.970i)5-s + (−0.386 + 1.68i)6-s + (−3.13 − 0.840i)7-s + (0.707 + 0.707i)8-s + (1.30 − 2.70i)9-s + (3.24 − 1.87i)10-s + (−1.54 − 1.54i)11-s + (−0.920 − 1.46i)12-s + (0.305 − 3.59i)13-s + (2.81 − 1.62i)14-s + (−6.20 + 1.90i)15-s − 1.00·16-s + (−3.50 + 6.07i)17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (0.847 − 0.531i)3-s − 0.500i·4-s + (−1.61 − 0.433i)5-s + (−0.157 + 0.689i)6-s + (−1.18 − 0.317i)7-s + (0.250 + 0.250i)8-s + (0.435 − 0.900i)9-s + (1.02 − 0.592i)10-s + (−0.466 − 0.466i)11-s + (−0.265 − 0.423i)12-s + (0.0847 − 0.996i)13-s + (0.751 − 0.434i)14-s + (−1.60 + 0.493i)15-s − 0.250·16-s + (−0.850 + 1.47i)17-s + ⋯ |
Λ(s)=(=(234s/2ΓC(s)L(s)(−0.398+0.917i)Λ(2−s)
Λ(s)=(=(234s/2ΓC(s+1/2)L(s)(−0.398+0.917i)Λ(1−s)
Degree: |
2 |
Conductor: |
234
= 2⋅32⋅13
|
Sign: |
−0.398+0.917i
|
Analytic conductor: |
1.86849 |
Root analytic conductor: |
1.36693 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ234(11,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 234, ( :1/2), −0.398+0.917i)
|
Particular Values
L(1) |
≈ |
0.323588−0.493123i |
L(21) |
≈ |
0.323588−0.493123i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.707−0.707i)T |
| 3 | 1+(−1.46+0.920i)T |
| 13 | 1+(−0.305+3.59i)T |
good | 5 | 1+(3.62+0.970i)T+(4.33+2.5i)T2 |
| 7 | 1+(3.13+0.840i)T+(6.06+3.5i)T2 |
| 11 | 1+(1.54+1.54i)T+11iT2 |
| 17 | 1+(3.50−6.07i)T+(−8.5−14.7i)T2 |
| 19 | 1+(−4.70+1.26i)T+(16.4−9.5i)T2 |
| 23 | 1+(−0.415+0.720i)T+(−11.5−19.9i)T2 |
| 29 | 1+9.26iT−29T2 |
| 31 | 1+(0.141−0.527i)T+(−26.8−15.5i)T2 |
| 37 | 1+(−1.90−0.509i)T+(32.0+18.5i)T2 |
| 41 | 1+(−0.455−1.70i)T+(−35.5+20.5i)T2 |
| 43 | 1+(−0.0503+0.0290i)T+(21.5−37.2i)T2 |
| 47 | 1+(−5.83+1.56i)T+(40.7−23.5i)T2 |
| 53 | 1+1.62iT−53T2 |
| 59 | 1+(−2.06−2.06i)T+59iT2 |
| 61 | 1+(1.78+3.10i)T+(−30.5+52.8i)T2 |
| 67 | 1+(12.5−3.36i)T+(58.0−33.5i)T2 |
| 71 | 1+(2.31+8.65i)T+(−61.4+35.5i)T2 |
| 73 | 1+(−3.38+3.38i)T−73iT2 |
| 79 | 1+(5.86−10.1i)T+(−39.5−68.4i)T2 |
| 83 | 1+(−3.75−14.0i)T+(−71.8+41.5i)T2 |
| 89 | 1+(−3.50+13.0i)T+(−77.0−44.5i)T2 |
| 97 | 1+(−2.55+9.53i)T+(−84.0−48.5i)T2 |
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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
−12.03074364160026922251905871478, −10.80415968259467608009502100858, −9.684016369733334097313495479945, −8.562019274508685340554084101128, −7.986918861537168558575839604106, −7.20082605575935112959413979843, −6.04954013551769667991339416734, −4.14342575215319929929669250670, −3.10403998994392163057218815516, −0.50446922619330655031139896254,
2.74427461422090755103101498769, 3.54607889338857463233865418916, 4.67711730111268291149035184041, 7.03786007603023722916131678376, 7.52947320134811740993530757409, 8.867220014798975447830145173163, 9.411651427954764093860918296860, 10.50932840702392159635766804887, 11.46235526604682834195138875459, 12.24099644076468710177521130630