L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.548 − 0.316i)5-s + (−2.15 + 1.24i)7-s − 0.999i·8-s − 0.633·10-s + (4.20 − 2.43i)11-s + (−0.541 − 3.56i)13-s + (−1.24 + 2.15i)14-s + (−0.5 − 0.866i)16-s + 6.27·17-s − 4.86i·19-s + (−0.548 + 0.316i)20-s + (2.43 − 4.20i)22-s + (1.77 − 3.07i)23-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.245 − 0.141i)5-s + (−0.814 + 0.469i)7-s − 0.353i·8-s − 0.200·10-s + (1.26 − 0.732i)11-s + (−0.150 − 0.988i)13-s + (−0.332 + 0.575i)14-s + (−0.125 − 0.216i)16-s + 1.52·17-s − 1.11i·19-s + (−0.122 + 0.0707i)20-s + (0.518 − 0.897i)22-s + (0.369 − 0.640i)23-s + ⋯ |
Λ(s)=(=(702s/2ΓC(s)L(s)(0.162+0.986i)Λ(2−s)
Λ(s)=(=(702s/2ΓC(s+1/2)L(s)(0.162+0.986i)Λ(1−s)
Degree: |
2 |
Conductor: |
702
= 2⋅33⋅13
|
Sign: |
0.162+0.986i
|
Analytic conductor: |
5.60549 |
Root analytic conductor: |
2.36759 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ702(415,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 702, ( :1/2), 0.162+0.986i)
|
Particular Values
L(1) |
≈ |
1.47623−1.25291i |
L(21) |
≈ |
1.47623−1.25291i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.866+0.5i)T |
| 3 | 1 |
| 13 | 1+(0.541+3.56i)T |
good | 5 | 1+(0.548+0.316i)T+(2.5+4.33i)T2 |
| 7 | 1+(2.15−1.24i)T+(3.5−6.06i)T2 |
| 11 | 1+(−4.20+2.43i)T+(5.5−9.52i)T2 |
| 17 | 1−6.27T+17T2 |
| 19 | 1+4.86iT−19T2 |
| 23 | 1+(−1.77+3.07i)T+(−11.5−19.9i)T2 |
| 29 | 1+(0.415+0.719i)T+(−14.5+25.1i)T2 |
| 31 | 1+(3.73+2.15i)T+(15.5+26.8i)T2 |
| 37 | 1−7.81iT−37T2 |
| 41 | 1+(0.0678+0.0391i)T+(20.5+35.5i)T2 |
| 43 | 1+(−4.84−8.38i)T+(−21.5+37.2i)T2 |
| 47 | 1+(4.30−2.48i)T+(23.5−40.7i)T2 |
| 53 | 1−6.36T+53T2 |
| 59 | 1+(7.86+4.54i)T+(29.5+51.0i)T2 |
| 61 | 1+(−5.28−9.15i)T+(−30.5+52.8i)T2 |
| 67 | 1+(6.82+3.94i)T+(33.5+58.0i)T2 |
| 71 | 1−12.3iT−71T2 |
| 73 | 1−1.05iT−73T2 |
| 79 | 1+(1.68+2.92i)T+(−39.5+68.4i)T2 |
| 83 | 1+(−13.1+7.60i)T+(41.5−71.8i)T2 |
| 89 | 1−0.595iT−89T2 |
| 97 | 1+(−14.7+8.53i)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
−10.26331402986951499253431146845, −9.515531328515645836578670183629, −8.655431965633763642705592385059, −7.58395547533545642482443238251, −6.39181992999191738383081342554, −5.83073187077393233607323500043, −4.67913358365108459611498157334, −3.49940218418819399643182140858, −2.79962736618382972204038907440, −0.906675024855119968353494345099,
1.70554765704910145424692881547, 3.58996390089658055176987467963, 3.85544842734360900968435894677, 5.27398570778197961899356119485, 6.25153863588606405221994895063, 7.13468926968142168914653287255, 7.61494964483184945760017327750, 9.078298235585259124266197562395, 9.665645486997423547127614464289, 10.64670372975041778476599068400