L(s) = 1 | + (0.846 − 1.13i)2-s + (0.635 + 0.797i)3-s + (−0.567 − 1.91i)4-s + (−0.726 + 0.579i)5-s + (1.44 − 0.0455i)6-s + (2.60 − 0.472i)7-s + (−2.65 − 0.980i)8-s + (0.436 − 1.91i)9-s + (0.0415 + 1.31i)10-s + (5.04 − 1.15i)11-s + (1.16 − 1.67i)12-s + (−2.71 + 0.618i)13-s + (1.66 − 3.34i)14-s + (−0.923 − 0.210i)15-s + (−3.35 + 2.17i)16-s + (−2.67 + 5.55i)17-s + ⋯ |
L(s) = 1 | + (0.598 − 0.801i)2-s + (0.367 + 0.460i)3-s + (−0.283 − 0.958i)4-s + (−0.324 + 0.259i)5-s + (0.588 − 0.0185i)6-s + (0.983 − 0.178i)7-s + (−0.937 − 0.346i)8-s + (0.145 − 0.637i)9-s + (0.0131 + 0.415i)10-s + (1.52 − 0.347i)11-s + (0.337 − 0.482i)12-s + (−0.751 + 0.171i)13-s + (0.445 − 0.895i)14-s + (−0.238 − 0.0544i)15-s + (−0.839 + 0.544i)16-s + (−0.648 + 1.34i)17-s + ⋯ |
Λ(s)=(=(196s/2ΓC(s)L(s)(0.576+0.817i)Λ(2−s)
Λ(s)=(=(196s/2ΓC(s+1/2)L(s)(0.576+0.817i)Λ(1−s)
Degree: |
2 |
Conductor: |
196
= 22⋅72
|
Sign: |
0.576+0.817i
|
Analytic conductor: |
1.56506 |
Root analytic conductor: |
1.25102 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ196(111,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 196, ( :1/2), 0.576+0.817i)
|
Particular Values
L(1) |
≈ |
1.54232−0.799358i |
L(21) |
≈ |
1.54232−0.799358i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.846+1.13i)T |
| 7 | 1+(−2.60+0.472i)T |
good | 3 | 1+(−0.635−0.797i)T+(−0.667+2.92i)T2 |
| 5 | 1+(0.726−0.579i)T+(1.11−4.87i)T2 |
| 11 | 1+(−5.04+1.15i)T+(9.91−4.77i)T2 |
| 13 | 1+(2.71−0.618i)T+(11.7−5.64i)T2 |
| 17 | 1+(2.67−5.55i)T+(−10.5−13.2i)T2 |
| 19 | 1+5.31T+19T2 |
| 23 | 1+(−2.67−5.54i)T+(−14.3+17.9i)T2 |
| 29 | 1+(6.99+3.37i)T+(18.0+22.6i)T2 |
| 31 | 1+2.80T+31T2 |
| 37 | 1+(5.03+2.42i)T+(23.0+28.9i)T2 |
| 41 | 1+(−8.64+6.89i)T+(9.12−39.9i)T2 |
| 43 | 1+(1.76+1.40i)T+(9.56+41.9i)T2 |
| 47 | 1+(−1.31−5.75i)T+(−42.3+20.3i)T2 |
| 53 | 1+(−3.53+1.70i)T+(33.0−41.4i)T2 |
| 59 | 1+(−3.63+4.55i)T+(−13.1−57.5i)T2 |
| 61 | 1+(2.83−5.89i)T+(−38.0−47.6i)T2 |
| 67 | 1−7.05iT−67T2 |
| 71 | 1+(−0.607−1.26i)T+(−44.2+55.5i)T2 |
| 73 | 1+(4.23+0.966i)T+(65.7+31.6i)T2 |
| 79 | 1+1.47iT−79T2 |
| 83 | 1+(−2.36+10.3i)T+(−74.7−36.0i)T2 |
| 89 | 1+(−8.05−1.83i)T+(80.1+38.6i)T2 |
| 97 | 1+11.4iT−97T2 |
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
−12.20443655964007949230473200448, −11.35797075459324944568552559663, −10.69519703535455199803634318293, −9.389955223287259114032485734001, −8.768485943450761848970324210022, −7.09357114992357455551968840252, −5.81169388962977913837357046484, −4.20753451333743895641702717382, −3.74902822572673252686977171572, −1.78830585698408429409879593902,
2.31381227817840939686441650973, 4.28702565813818713758929106904, 5.00188081947156964901612339771, 6.65703701427684333005423272850, 7.43215323702893397760030703346, 8.430215747507871039332508850027, 9.176606110548302843275451031845, 11.01415079559243791452776981793, 12.00463185335235171244770109469, 12.71326116579253336890074977013