L(s) = 1 | + (−0.707 + 0.707i)2-s + (−1.57 + 0.726i)3-s − 1.00i·4-s + (0.653 + 0.175i)5-s + (0.597 − 1.62i)6-s + (−3.90 − 1.04i)7-s + (0.707 + 0.707i)8-s + (1.94 − 2.28i)9-s + (−0.585 + 0.338i)10-s + (−0.502 − 0.502i)11-s + (0.726 + 1.57i)12-s + (−2.29 − 2.77i)13-s + (3.49 − 2.01i)14-s + (−1.15 + 0.199i)15-s − 1.00·16-s + (3.24 − 5.61i)17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (−0.907 + 0.419i)3-s − 0.500i·4-s + (0.292 + 0.0782i)5-s + (0.243 − 0.663i)6-s + (−1.47 − 0.394i)7-s + (0.250 + 0.250i)8-s + (0.647 − 0.761i)9-s + (−0.185 + 0.106i)10-s + (−0.151 − 0.151i)11-s + (0.209 + 0.453i)12-s + (−0.637 − 0.770i)13-s + (0.934 − 0.539i)14-s + (−0.297 + 0.0515i)15-s − 0.250·16-s + (0.786 − 1.36i)17-s + ⋯ |
Λ(s)=(=(234s/2ΓC(s)L(s)(0.112+0.993i)Λ(2−s)
Λ(s)=(=(234s/2ΓC(s+1/2)L(s)(0.112+0.993i)Λ(1−s)
Degree: |
2 |
Conductor: |
234
= 2⋅32⋅13
|
Sign: |
0.112+0.993i
|
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), 0.112+0.993i)
|
Particular Values
L(1) |
≈ |
0.250040−0.223345i |
L(21) |
≈ |
0.250040−0.223345i |
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+(1.57−0.726i)T |
| 13 | 1+(2.29+2.77i)T |
good | 5 | 1+(−0.653−0.175i)T+(4.33+2.5i)T2 |
| 7 | 1+(3.90+1.04i)T+(6.06+3.5i)T2 |
| 11 | 1+(0.502+0.502i)T+11iT2 |
| 17 | 1+(−3.24+5.61i)T+(−8.5−14.7i)T2 |
| 19 | 1+(0.253−0.0679i)T+(16.4−9.5i)T2 |
| 23 | 1+(−0.860+1.49i)T+(−11.5−19.9i)T2 |
| 29 | 1+1.28iT−29T2 |
| 31 | 1+(−1.04+3.89i)T+(−26.8−15.5i)T2 |
| 37 | 1+(7.96+2.13i)T+(32.0+18.5i)T2 |
| 41 | 1+(−2.39−8.94i)T+(−35.5+20.5i)T2 |
| 43 | 1+(5.67−3.27i)T+(21.5−37.2i)T2 |
| 47 | 1+(9.07−2.43i)T+(40.7−23.5i)T2 |
| 53 | 1−6.34iT−53T2 |
| 59 | 1+(3.52+3.52i)T+59iT2 |
| 61 | 1+(1.64+2.85i)T+(−30.5+52.8i)T2 |
| 67 | 1+(−10.0+2.70i)T+(58.0−33.5i)T2 |
| 71 | 1+(2.09+7.82i)T+(−61.4+35.5i)T2 |
| 73 | 1+(1.40−1.40i)T−73iT2 |
| 79 | 1+(−7.29+12.6i)T+(−39.5−68.4i)T2 |
| 83 | 1+(−1.04−3.90i)T+(−71.8+41.5i)T2 |
| 89 | 1+(2.64−9.85i)T+(−77.0−44.5i)T2 |
| 97 | 1+(−0.520+1.94i)T+(−84.0−48.5i)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
−11.89525451707774612525624335150, −10.69794958110887916538089510891, −9.755552471637068486689899489360, −9.591164822935936238823358703334, −7.79249875025329875504804660060, −6.74668856679165901609556055885, −5.96303974874878413972181553434, −4.88162305254278628553277744137, −3.19257265030370646425806116992, −0.34570584695317748369170126342,
1.88096976251078727627822718868, 3.58901930326341224334228308131, 5.32195317498869915273335430380, 6.41056625244526405319067680414, 7.25333292911417480518355737318, 8.658543761590445363589947704119, 9.850453201472966548107026419473, 10.26816080993993132451837487572, 11.55310210325390548217440494883, 12.40925791854938451744987758951