L(s) = 1 | + (0.707 − 0.707i)2-s + (−1.25 − 1.19i)3-s − 1.00i·4-s + (−0.670 − 0.179i)5-s + (−1.73 + 0.0409i)6-s + (−4.43 − 1.18i)7-s + (−0.707 − 0.707i)8-s + (0.141 + 2.99i)9-s + (−0.601 + 0.347i)10-s + (−1.59 − 1.59i)11-s + (−1.19 + 1.25i)12-s + (3.60 + 0.0239i)13-s + (−3.97 + 2.29i)14-s + (0.625 + 1.02i)15-s − 1.00·16-s + (2.32 − 4.02i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.723 − 0.690i)3-s − 0.500i·4-s + (−0.299 − 0.0803i)5-s + (−0.706 + 0.0167i)6-s + (−1.67 − 0.449i)7-s + (−0.250 − 0.250i)8-s + (0.0473 + 0.998i)9-s + (−0.190 + 0.109i)10-s + (−0.479 − 0.479i)11-s + (−0.345 + 0.361i)12-s + (0.999 + 0.00665i)13-s + (−1.06 + 0.613i)14-s + (0.161 + 0.265i)15-s − 0.250·16-s + (0.563 − 0.975i)17-s + ⋯ |
Λ(s)=(=(234s/2ΓC(s)L(s)(−0.986+0.165i)Λ(2−s)
Λ(s)=(=(234s/2ΓC(s+1/2)L(s)(−0.986+0.165i)Λ(1−s)
Degree: |
2 |
Conductor: |
234
= 2⋅32⋅13
|
Sign: |
−0.986+0.165i
|
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), −0.986+0.165i)
|
Particular Values
L(1) |
≈ |
0.0629357−0.753915i |
L(21) |
≈ |
0.0629357−0.753915i |
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+(1.25+1.19i)T |
| 13 | 1+(−3.60−0.0239i)T |
good | 5 | 1+(0.670+0.179i)T+(4.33+2.5i)T2 |
| 7 | 1+(4.43+1.18i)T+(6.06+3.5i)T2 |
| 11 | 1+(1.59+1.59i)T+11iT2 |
| 17 | 1+(−2.32+4.02i)T+(−8.5−14.7i)T2 |
| 19 | 1+(2.81−0.753i)T+(16.4−9.5i)T2 |
| 23 | 1+(−2.37+4.10i)T+(−11.5−19.9i)T2 |
| 29 | 1+4.88iT−29T2 |
| 31 | 1+(1.74−6.51i)T+(−26.8−15.5i)T2 |
| 37 | 1+(−3.44−0.924i)T+(32.0+18.5i)T2 |
| 41 | 1+(1.48+5.54i)T+(−35.5+20.5i)T2 |
| 43 | 1+(−4.59+2.65i)T+(21.5−37.2i)T2 |
| 47 | 1+(−11.6+3.12i)T+(40.7−23.5i)T2 |
| 53 | 1−5.06iT−53T2 |
| 59 | 1+(−3.79−3.79i)T+59iT2 |
| 61 | 1+(1.56+2.70i)T+(−30.5+52.8i)T2 |
| 67 | 1+(12.6−3.38i)T+(58.0−33.5i)T2 |
| 71 | 1+(4.16+15.5i)T+(−61.4+35.5i)T2 |
| 73 | 1+(5.55−5.55i)T−73iT2 |
| 79 | 1+(1.55−2.69i)T+(−39.5−68.4i)T2 |
| 83 | 1+(−2.45−9.17i)T+(−71.8+41.5i)T2 |
| 89 | 1+(0.648−2.41i)T+(−77.0−44.5i)T2 |
| 97 | 1+(0.317−1.18i)T+(−84.0−48.5i)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
−11.96882066816761200640942075546, −10.79534639494219081444004614593, −10.23316030943464985236522109899, −8.868881636025087188413331922286, −7.43210069128308960580559921666, −6.41657287840296855911874684157, −5.66632252230353468995689670788, −4.09263967220680990316266650833, −2.79855920146979738037193000879, −0.56410138344717579706925277596,
3.24420077702307549269926172752, 4.13070787457407539071603550402, 5.70711127276821269586425292053, 6.18297186000418104276401650558, 7.38124794852982555505405861556, 8.870657936626562857157593481008, 9.751376553025913507387061790114, 10.74192573937188802771206888837, 11.77648176326977736820218311168, 12.81809831417583531794254530688