L(s) = 1 | + (0.707 − 0.707i)2-s + (−1.36 + 1.06i)3-s − 1.00i·4-s + (−3.36 − 0.901i)5-s + (−0.211 + 1.71i)6-s + (0.0541 + 0.0145i)7-s + (−0.707 − 0.707i)8-s + (0.727 − 2.91i)9-s + (−3.01 + 1.74i)10-s + (−3.65 − 3.65i)11-s + (1.06 + 1.36i)12-s + (−3.37 + 1.25i)13-s + (0.0485 − 0.0280i)14-s + (5.55 − 2.35i)15-s − 1.00·16-s + (−1.83 + 3.17i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.788 + 0.615i)3-s − 0.500i·4-s + (−1.50 − 0.403i)5-s + (−0.0863 + 0.701i)6-s + (0.0204 + 0.00548i)7-s + (−0.250 − 0.250i)8-s + (0.242 − 0.970i)9-s + (−0.954 + 0.550i)10-s + (−1.10 − 1.10i)11-s + (0.307 + 0.394i)12-s + (−0.937 + 0.348i)13-s + (0.0129 − 0.00749i)14-s + (1.43 − 0.608i)15-s − 0.250·16-s + (−0.443 + 0.768i)17-s + ⋯ |
Λ(s)=(=(234s/2ΓC(s)L(s)(−0.972+0.234i)Λ(2−s)
Λ(s)=(=(234s/2ΓC(s+1/2)L(s)(−0.972+0.234i)Λ(1−s)
Degree: |
2 |
Conductor: |
234
= 2⋅32⋅13
|
Sign: |
−0.972+0.234i
|
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), −0.972+0.234i)
|
Particular Values
L(1) |
≈ |
0.0421597−0.354394i |
L(21) |
≈ |
0.0421597−0.354394i |
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+(1.36−1.06i)T |
| 13 | 1+(3.37−1.25i)T |
good | 5 | 1+(3.36+0.901i)T+(4.33+2.5i)T2 |
| 7 | 1+(−0.0541−0.0145i)T+(6.06+3.5i)T2 |
| 11 | 1+(3.65+3.65i)T+11iT2 |
| 17 | 1+(1.83−3.17i)T+(−8.5−14.7i)T2 |
| 19 | 1+(−0.677+0.181i)T+(16.4−9.5i)T2 |
| 23 | 1+(−2.89+5.01i)T+(−11.5−19.9i)T2 |
| 29 | 1−3.45iT−29T2 |
| 31 | 1+(−1.54+5.75i)T+(−26.8−15.5i)T2 |
| 37 | 1+(−7.28−1.95i)T+(32.0+18.5i)T2 |
| 41 | 1+(0.780+2.91i)T+(−35.5+20.5i)T2 |
| 43 | 1+(4.70−2.71i)T+(21.5−37.2i)T2 |
| 47 | 1+(4.82−1.29i)T+(40.7−23.5i)T2 |
| 53 | 1+3.89iT−53T2 |
| 59 | 1+(−0.745−0.745i)T+59iT2 |
| 61 | 1+(−1.50−2.60i)T+(−30.5+52.8i)T2 |
| 67 | 1+(10.5−2.81i)T+(58.0−33.5i)T2 |
| 71 | 1+(3.98+14.8i)T+(−61.4+35.5i)T2 |
| 73 | 1+(−5.86+5.86i)T−73iT2 |
| 79 | 1+(−2.76+4.79i)T+(−39.5−68.4i)T2 |
| 83 | 1+(4.13+15.4i)T+(−71.8+41.5i)T2 |
| 89 | 1+(4.41−16.4i)T+(−77.0−44.5i)T2 |
| 97 | 1+(0.595−2.22i)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
−11.60516618092853872925255285108, −11.06647162430636886276903175600, −10.19161635545162009487479391910, −8.855983949840098493318263809724, −7.80454270475438437600529317303, −6.39003151798883363574597819336, −5.06119630004799371695619801222, −4.35258276183072364261775650455, −3.15964905747615660937125753374, −0.26178179756609240541533261293,
2.79390830393581665291141121950, 4.50979577668567553296086977940, 5.25345625128837565431112852757, 6.86635007375612208206516326800, 7.45157781708551365029225632341, 8.045576306464928065793494564088, 9.905959479798266203450587556796, 11.11881057815362785423547240306, 11.78485630885711604546190863360, 12.56424586291609200227499125652