L(s) = 1 | + 4i·5-s + (−3 − 2i)13-s + 2·17-s − 11·25-s − 10·29-s − 12i·37-s + 8i·41-s + 7·49-s − 14·53-s − 10·61-s + (8 − 12i)65-s + 16i·73-s + 8i·85-s − 16i·89-s − 8i·97-s + ⋯ |
L(s) = 1 | + 1.78i·5-s + (−0.832 − 0.554i)13-s + 0.485·17-s − 2.20·25-s − 1.85·29-s − 1.97i·37-s + 1.24i·41-s + 49-s − 1.92·53-s − 1.28·61-s + (0.992 − 1.48i)65-s + 1.87i·73-s + 0.867i·85-s − 1.69i·89-s − 0.812i·97-s + ⋯ |
Λ(s)=(=(3744s/2ΓC(s)L(s)(−0.554+0.832i)Λ(2−s)
Λ(s)=(=(3744s/2ΓC(s+1/2)L(s)(−0.554+0.832i)Λ(1−s)
Degree: |
2 |
Conductor: |
3744
= 25⋅32⋅13
|
Sign: |
−0.554+0.832i
|
Analytic conductor: |
29.8959 |
Root analytic conductor: |
5.46772 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ3744(3457,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
1
|
Selberg data: |
(2, 3744, ( :1/2), −0.554+0.832i)
|
Particular Values
L(1) |
= |
0 |
L(21) |
= |
0 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 13 | 1+(3+2i)T |
good | 5 | 1−4iT−5T2 |
| 7 | 1−7T2 |
| 11 | 1−11T2 |
| 17 | 1−2T+17T2 |
| 19 | 1−19T2 |
| 23 | 1+23T2 |
| 29 | 1+10T+29T2 |
| 31 | 1−31T2 |
| 37 | 1+12iT−37T2 |
| 41 | 1−8iT−41T2 |
| 43 | 1+43T2 |
| 47 | 1−47T2 |
| 53 | 1+14T+53T2 |
| 59 | 1−59T2 |
| 61 | 1+10T+61T2 |
| 67 | 1−67T2 |
| 71 | 1−71T2 |
| 73 | 1−16iT−73T2 |
| 79 | 1+79T2 |
| 83 | 1−83T2 |
| 89 | 1+16iT−89T2 |
| 97 | 1+8iT−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
−7.920250557861090988000720899088, −7.47349600087989102088205715604, −6.94345609829164421040776940499, −6.00867871883172260252049158466, −5.50262499783384054777244257763, −4.27532854977663050671091326032, −3.38347913481718259752106087415, −2.77348086636941653821329988722, −1.87648676244566357541774297229, 0,
1.26214535222265802680455156179, 2.09023021001683883419116091656, 3.45790316511305003191858201870, 4.36913661305886093985718531011, 4.98250105396219012433013251598, 5.56776528716580331474882766751, 6.47969997481039407164373511739, 7.57087221163524274035714087100, 7.973832741044919213494427292927