L(s) = 1 | + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (2.02 − 3.51i)5-s + (−0.376 − 1.40i)7-s + (0.707 − 0.707i)8-s + (1.04 − 3.91i)10-s + (−2.07 + 2.07i)11-s + (2.10 − 3.64i)13-s + (−0.728 − 1.26i)14-s + (0.500 − 0.866i)16-s + (2.30 + 0.617i)17-s + (−3.11 + 1.80i)19-s − 4.05i·20-s + (−1.46 + 2.54i)22-s + (4.74 + 4.74i)23-s + ⋯ |
L(s) = 1 | + (0.683 − 0.183i)2-s + (0.433 − 0.249i)4-s + (0.906 − 1.57i)5-s + (−0.142 − 0.531i)7-s + (0.249 − 0.249i)8-s + (0.331 − 1.23i)10-s + (−0.626 + 0.626i)11-s + (0.584 − 1.01i)13-s + (−0.194 − 0.337i)14-s + (0.125 − 0.216i)16-s + (0.558 + 0.149i)17-s + (−0.715 + 0.413i)19-s − 0.906i·20-s + (−0.313 + 0.542i)22-s + (0.989 + 0.989i)23-s + ⋯ |
Λ(s)=(=(1098s/2ΓC(s)L(s)(−0.123+0.992i)Λ(2−s)
Λ(s)=(=(1098s/2ΓC(s+1/2)L(s)(−0.123+0.992i)Λ(1−s)
Degree: |
2 |
Conductor: |
1098
= 2⋅32⋅61
|
Sign: |
−0.123+0.992i
|
Analytic conductor: |
8.76757 |
Root analytic conductor: |
2.96100 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1098(467,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1098, ( :1/2), −0.123+0.992i)
|
Particular Values
L(1) |
≈ |
2.727788990 |
L(21) |
≈ |
2.727788990 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.965+0.258i)T |
| 3 | 1 |
| 61 | 1+(−7.31−2.73i)T |
good | 5 | 1+(−2.02+3.51i)T+(−2.5−4.33i)T2 |
| 7 | 1+(0.376+1.40i)T+(−6.06+3.5i)T2 |
| 11 | 1+(2.07−2.07i)T−11iT2 |
| 13 | 1+(−2.10+3.64i)T+(−6.5−11.2i)T2 |
| 17 | 1+(−2.30−0.617i)T+(14.7+8.5i)T2 |
| 19 | 1+(3.11−1.80i)T+(9.5−16.4i)T2 |
| 23 | 1+(−4.74−4.74i)T+23iT2 |
| 29 | 1+(5.85+1.56i)T+(25.1+14.5i)T2 |
| 31 | 1+(0.0233−0.0871i)T+(−26.8−15.5i)T2 |
| 37 | 1+(−2.07−2.07i)T+37iT2 |
| 41 | 1+11.6T+41T2 |
| 43 | 1+(−4.34+1.16i)T+(37.2−21.5i)T2 |
| 47 | 1+(−2.19+1.26i)T+(23.5−40.7i)T2 |
| 53 | 1+(1.83+1.83i)T+53iT2 |
| 59 | 1+(−3.27−12.2i)T+(−51.0+29.5i)T2 |
| 67 | 1+(−5.71+1.53i)T+(58.0−33.5i)T2 |
| 71 | 1+(−8.88−2.38i)T+(61.4+35.5i)T2 |
| 73 | 1+(0.863+1.49i)T+(−36.5+63.2i)T2 |
| 79 | 1+(1.68+6.27i)T+(−68.4+39.5i)T2 |
| 83 | 1+(3.26+1.88i)T+(41.5+71.8i)T2 |
| 89 | 1+(13.3−13.3i)T−89iT2 |
| 97 | 1+(−9.18+5.30i)T+(48.5−84.0i)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
−9.878026682265753194797233696563, −8.829849129226760141618132991572, −8.046421047471175574464115141916, −7.06888913332312251926688887309, −5.74174258261306644267879263219, −5.43974277938961408961019387471, −4.50456221051529759184907493737, −3.49752547946673453837440911268, −2.05404714295386899282687792114, −0.994193829111731350084876661334,
2.06909773723131838195310790798, 2.83747001715280827838800226784, 3.71408318154853550421706034203, 5.13426397045171488736088943837, 5.94984869498114793101197211555, 6.59472486565498946303863426785, 7.19466468984692916001912857251, 8.418973979691715045908159555552, 9.315548559315554857813782665001, 10.24621021929270968741249610776