L(s) = 1 | + (−1.20 + 2.09i)3-s + (−1.91 − 3.31i)5-s + (−1 − 2.44i)7-s + (−1.41 − 2.44i)9-s + (−0.207 + 0.358i)11-s − 2.82·13-s + 9.24·15-s + (2.91 − 5.04i)17-s + (−1.79 − 3.10i)19-s + (6.32 + 0.866i)21-s + (−1.62 − 2.80i)23-s + (−4.82 + 8.36i)25-s − 0.414·27-s + 2.82·29-s + (−4.20 + 7.28i)31-s + ⋯ |
L(s) = 1 | + (−0.696 + 1.20i)3-s + (−0.856 − 1.48i)5-s + (−0.377 − 0.925i)7-s + (−0.471 − 0.816i)9-s + (−0.0624 + 0.108i)11-s − 0.784·13-s + 2.38·15-s + (0.706 − 1.22i)17-s + (−0.411 − 0.712i)19-s + (1.38 + 0.188i)21-s + (−0.338 − 0.585i)23-s + (−0.965 + 1.67i)25-s − 0.0797·27-s + 0.525·29-s + (−0.755 + 1.30i)31-s + ⋯ |
Λ(s)=(=(224s/2ΓC(s)L(s)(−0.198+0.980i)Λ(2−s)
Λ(s)=(=(224s/2ΓC(s+1/2)L(s)(−0.198+0.980i)Λ(1−s)
Degree: |
2 |
Conductor: |
224
= 25⋅7
|
Sign: |
−0.198+0.980i
|
Analytic conductor: |
1.78864 |
Root analytic conductor: |
1.33740 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ224(193,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 224, ( :1/2), −0.198+0.980i)
|
Particular Values
L(1) |
≈ |
0.298990−0.365447i |
L(21) |
≈ |
0.298990−0.365447i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 7 | 1+(1+2.44i)T |
good | 3 | 1+(1.20−2.09i)T+(−1.5−2.59i)T2 |
| 5 | 1+(1.91+3.31i)T+(−2.5+4.33i)T2 |
| 11 | 1+(0.207−0.358i)T+(−5.5−9.52i)T2 |
| 13 | 1+2.82T+13T2 |
| 17 | 1+(−2.91+5.04i)T+(−8.5−14.7i)T2 |
| 19 | 1+(1.79+3.10i)T+(−9.5+16.4i)T2 |
| 23 | 1+(1.62+2.80i)T+(−11.5+19.9i)T2 |
| 29 | 1−2.82T+29T2 |
| 31 | 1+(4.20−7.28i)T+(−15.5−26.8i)T2 |
| 37 | 1+(−1.32−2.30i)T+(−18.5+32.0i)T2 |
| 41 | 1+1.17T+41T2 |
| 43 | 1+1.65T+43T2 |
| 47 | 1+(3.79+6.56i)T+(−23.5+40.7i)T2 |
| 53 | 1+(−0.5+0.866i)T+(−26.5−45.8i)T2 |
| 59 | 1+(−4.44+7.70i)T+(−29.5−51.0i)T2 |
| 61 | 1+(−1.32−2.30i)T+(−30.5+52.8i)T2 |
| 67 | 1+(−5.62+9.73i)T+(−33.5−58.0i)T2 |
| 71 | 1−2.34T+71T2 |
| 73 | 1+(−1.67+2.89i)T+(−36.5−63.2i)T2 |
| 79 | 1+(−4.03−6.98i)T+(−39.5+68.4i)T2 |
| 83 | 1−15.3T+83T2 |
| 89 | 1+(−4.5−7.79i)T+(−44.5+77.0i)T2 |
| 97 | 1+6.82T+97T2 |
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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
−11.95447090380714967130093443241, −11.00470647595595192609309142053, −9.998302211544781354820083614034, −9.294811313398675576747381963330, −8.115216830677667084266445815261, −6.94125238705006727806446586258, −5.08899489121680019822125765024, −4.74970796429945316308409696946, −3.63476503124581709168269301517, −0.41493240246859933979110567976,
2.25281250701362989151535054890, 3.65352756788188829622888871893, 5.76986169724734378208272622338, 6.42665345028336530833284235642, 7.43686613639004446020035357095, 8.110494502140131984945290582221, 9.831682518992228136269149994960, 10.87432782568891830810023882521, 11.78212290915825815982425658793, 12.28018101645004293635632017827