L(s) = 1 | + (−2.38 + 1.99i)3-s + (0.939 + 0.342i)5-s + (2.42 − 4.19i)7-s + (1.15 − 6.55i)9-s + (−0.912 − 1.57i)11-s + (4.37 + 3.67i)13-s + (−2.91 + 1.06i)15-s + (0.843 + 4.78i)17-s + (3.11 − 3.04i)19-s + (2.61 + 14.8i)21-s + (−3.49 + 1.27i)23-s + (0.766 + 0.642i)25-s + (5.67 + 9.83i)27-s + (0.509 − 2.88i)29-s + (−0.598 + 1.03i)31-s + ⋯ |
L(s) = 1 | + (−1.37 + 1.15i)3-s + (0.420 + 0.152i)5-s + (0.915 − 1.58i)7-s + (0.385 − 2.18i)9-s + (−0.275 − 0.476i)11-s + (1.21 + 1.01i)13-s + (−0.753 + 0.274i)15-s + (0.204 + 1.16i)17-s + (0.715 − 0.698i)19-s + (0.570 + 3.23i)21-s + (−0.728 + 0.264i)23-s + (0.153 + 0.128i)25-s + (1.09 + 1.89i)27-s + (0.0945 − 0.536i)29-s + (−0.107 + 0.186i)31-s + ⋯ |
Λ(s)=(=(380s/2ΓC(s)L(s)(0.904−0.426i)Λ(2−s)
Λ(s)=(=(380s/2ΓC(s+1/2)L(s)(0.904−0.426i)Λ(1−s)
Degree: |
2 |
Conductor: |
380
= 22⋅5⋅19
|
Sign: |
0.904−0.426i
|
Analytic conductor: |
3.03431 |
Root analytic conductor: |
1.74192 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ380(161,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 380, ( :1/2), 0.904−0.426i)
|
Particular Values
L(1) |
≈ |
1.03357+0.231347i |
L(21) |
≈ |
1.03357+0.231347i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 5 | 1+(−0.939−0.342i)T |
| 19 | 1+(−3.11+3.04i)T |
good | 3 | 1+(2.38−1.99i)T+(0.520−2.95i)T2 |
| 7 | 1+(−2.42+4.19i)T+(−3.5−6.06i)T2 |
| 11 | 1+(0.912+1.57i)T+(−5.5+9.52i)T2 |
| 13 | 1+(−4.37−3.67i)T+(2.25+12.8i)T2 |
| 17 | 1+(−0.843−4.78i)T+(−15.9+5.81i)T2 |
| 23 | 1+(3.49−1.27i)T+(17.6−14.7i)T2 |
| 29 | 1+(−0.509+2.88i)T+(−27.2−9.91i)T2 |
| 31 | 1+(0.598−1.03i)T+(−15.5−26.8i)T2 |
| 37 | 1−3.79T+37T2 |
| 41 | 1+(−6.35+5.33i)T+(7.11−40.3i)T2 |
| 43 | 1+(−9.07−3.30i)T+(32.9+27.6i)T2 |
| 47 | 1+(−0.728+4.12i)T+(−44.1−16.0i)T2 |
| 53 | 1+(1.62−0.590i)T+(40.6−34.0i)T2 |
| 59 | 1+(−1.96−11.1i)T+(−55.4+20.1i)T2 |
| 61 | 1+(−1.03+0.377i)T+(46.7−39.2i)T2 |
| 67 | 1+(0.781−4.43i)T+(−62.9−22.9i)T2 |
| 71 | 1+(6.95+2.53i)T+(54.3+45.6i)T2 |
| 73 | 1+(5.48−4.60i)T+(12.6−71.8i)T2 |
| 79 | 1+(−2.27+1.90i)T+(13.7−77.7i)T2 |
| 83 | 1+(−5.31+9.20i)T+(−41.5−71.8i)T2 |
| 89 | 1+(−1.63−1.37i)T+(15.4+87.6i)T2 |
| 97 | 1+(1.46+8.30i)T+(−91.1+33.1i)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
−11.09177422970743790775679918854, −10.71698273796814326738912858039, −9.980638563810413139358736182926, −8.887883863077098615001608425155, −7.53360482306108254343067350432, −6.35157908341489345120051325477, −5.60229958346992836299760511786, −4.35690901188200451279198574283, −3.87467113004121336860686159177, −1.12117292142370320334710521878,
1.25054334959559957789252632942, 2.48471248123406263251746602988, 4.95318115746292254221754387691, 5.65384085946100672880480675844, 6.15442054851177742452176463731, 7.57595327515206972914889500530, 8.219684641069664931413429674260, 9.481214474950452303174196163941, 10.77829633268014740790740902780, 11.43327811372038136305097453001