L(s) = 1 | + (−0.5 − 0.866i)3-s + 4.14i·5-s + (2.08 + 1.20i)7-s + (−0.499 + 0.866i)9-s + (3 − 1.73i)11-s + (1.5 − 3.27i)13-s + (3.58 − 2.07i)15-s + (−2.58 + 4.48i)17-s + (−3 − 1.73i)19-s − 2.41i·21-s + (1 + 1.73i)23-s − 12.1·25-s + 0.999·27-s + (−1.58 − 2.75i)29-s + 1.05i·31-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + 1.85i·5-s + (0.789 + 0.455i)7-s + (−0.166 + 0.288i)9-s + (0.904 − 0.522i)11-s + (0.416 − 0.909i)13-s + (0.926 − 0.535i)15-s + (−0.628 + 1.08i)17-s + (−0.688 − 0.397i)19-s − 0.526i·21-s + (0.208 + 0.361i)23-s − 2.43·25-s + 0.192·27-s + (−0.295 − 0.511i)29-s + 0.188i·31-s + ⋯ |
Λ(s)=(=(156s/2ΓC(s)L(s)(0.822−0.569i)Λ(2−s)
Λ(s)=(=(156s/2ΓC(s+1/2)L(s)(0.822−0.569i)Λ(1−s)
Degree: |
2 |
Conductor: |
156
= 22⋅3⋅13
|
Sign: |
0.822−0.569i
|
Analytic conductor: |
1.24566 |
Root analytic conductor: |
1.11609 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ156(49,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 156, ( :1/2), 0.822−0.569i)
|
Particular Values
L(1) |
≈ |
1.05690+0.330012i |
L(21) |
≈ |
1.05690+0.330012i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1+(0.5+0.866i)T |
| 13 | 1+(−1.5+3.27i)T |
good | 5 | 1−4.14iT−5T2 |
| 7 | 1+(−2.08−1.20i)T+(3.5+6.06i)T2 |
| 11 | 1+(−3+1.73i)T+(5.5−9.52i)T2 |
| 17 | 1+(2.58−4.48i)T+(−8.5−14.7i)T2 |
| 19 | 1+(3+1.73i)T+(9.5+16.4i)T2 |
| 23 | 1+(−1−1.73i)T+(−11.5+19.9i)T2 |
| 29 | 1+(1.58+2.75i)T+(−14.5+25.1i)T2 |
| 31 | 1−1.05iT−31T2 |
| 37 | 1+(−6.58+3.80i)T+(18.5−32.0i)T2 |
| 41 | 1+(0.589−0.340i)T+(20.5−35.5i)T2 |
| 43 | 1+(−6.08+10.5i)T+(−21.5−37.2i)T2 |
| 47 | 1+10.3iT−47T2 |
| 53 | 1−1.17T+53T2 |
| 59 | 1+(10.1+5.87i)T+(29.5+51.0i)T2 |
| 61 | 1+(2.5−4.33i)T+(−30.5−52.8i)T2 |
| 67 | 1+(2.08−1.20i)T+(33.5−58.0i)T2 |
| 71 | 1+(3+1.73i)T+(35.5+61.4i)T2 |
| 73 | 1−14.8iT−73T2 |
| 79 | 1−1.82T+79T2 |
| 83 | 1+1.36iT−83T2 |
| 89 | 1+(−6+3.46i)T+(44.5−77.0i)T2 |
| 97 | 1+(−15.2−8.81i)T+(48.5+84.0i)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
−13.11306510884583046859183370332, −11.75164542981706926327522416233, −11.05940989265694986908398089580, −10.43548644321582412687364247619, −8.756441277280242303651446807199, −7.66014644328454915840600066703, −6.56059910586342399099871911475, −5.80147878610528880964109295369, −3.73982971399052110874871265241, −2.23080141833620993159800757180,
1.38926026282112266105857102187, 4.47374449864779133404461512491, 4.56106484069067626908446634377, 6.21014048053753204435227001131, 7.80214958935753081860106907376, 9.013706066831028281843662931595, 9.432009154217241701066443891018, 11.03862328200314933896035232560, 11.83246236524359221173862133024, 12.72404937857673455885675239814