L(s) = 1 | + (−0.493 + 1.32i)2-s + (1.01 − 0.419i)3-s + (−1.51 − 1.30i)4-s + (−1.26 + 3.06i)5-s + (0.0562 + 1.54i)6-s + (−0.757 − 0.757i)7-s + (2.48 − 1.35i)8-s + (−1.27 + 1.27i)9-s + (−3.43 − 3.19i)10-s + (−5.35 − 2.21i)11-s + (−2.08 − 0.690i)12-s + (0.382 + 0.923i)13-s + (1.37 − 0.629i)14-s + 3.63i·15-s + (0.578 + 3.95i)16-s + 4.42i·17-s + ⋯ |
L(s) = 1 | + (−0.348 + 0.937i)2-s + (0.584 − 0.242i)3-s + (−0.756 − 0.653i)4-s + (−0.566 + 1.36i)5-s + (0.0229 + 0.632i)6-s + (−0.286 − 0.286i)7-s + (0.876 − 0.480i)8-s + (−0.423 + 0.423i)9-s + (−1.08 − 1.00i)10-s + (−1.61 − 0.668i)11-s + (−0.600 − 0.199i)12-s + (0.106 + 0.256i)13-s + (0.367 − 0.168i)14-s + 0.937i·15-s + (0.144 + 0.989i)16-s + 1.07i·17-s + ⋯ |
Λ(s)=(=(416s/2ΓC(s)L(s)(−0.929+0.368i)Λ(2−s)
Λ(s)=(=(416s/2ΓC(s+1/2)L(s)(−0.929+0.368i)Λ(1−s)
Degree: |
2 |
Conductor: |
416
= 25⋅13
|
Sign: |
−0.929+0.368i
|
Analytic conductor: |
3.32177 |
Root analytic conductor: |
1.82257 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ416(53,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 416, ( :1/2), −0.929+0.368i)
|
Particular Values
L(1) |
≈ |
0.0879250−0.459892i |
L(21) |
≈ |
0.0879250−0.459892i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.493−1.32i)T |
| 13 | 1+(−0.382−0.923i)T |
good | 3 | 1+(−1.01+0.419i)T+(2.12−2.12i)T2 |
| 5 | 1+(1.26−3.06i)T+(−3.53−3.53i)T2 |
| 7 | 1+(0.757+0.757i)T+7iT2 |
| 11 | 1+(5.35+2.21i)T+(7.77+7.77i)T2 |
| 17 | 1−4.42iT−17T2 |
| 19 | 1+(1.84+4.46i)T+(−13.4+13.4i)T2 |
| 23 | 1+(2.30−2.30i)T−23iT2 |
| 29 | 1+(−4.83+2.00i)T+(20.5−20.5i)T2 |
| 31 | 1+2.03T+31T2 |
| 37 | 1+(1.25−3.03i)T+(−26.1−26.1i)T2 |
| 41 | 1+(7.20−7.20i)T−41iT2 |
| 43 | 1+(2.37+0.981i)T+(30.4+30.4i)T2 |
| 47 | 1−3.59iT−47T2 |
| 53 | 1+(−9.03−3.74i)T+(37.4+37.4i)T2 |
| 59 | 1+(−1.81+4.37i)T+(−41.7−41.7i)T2 |
| 61 | 1+(−4.32+1.79i)T+(43.1−43.1i)T2 |
| 67 | 1+(−6.15+2.54i)T+(47.3−47.3i)T2 |
| 71 | 1+(−5.53−5.53i)T+71iT2 |
| 73 | 1+(8.59−8.59i)T−73iT2 |
| 79 | 1−7.19iT−79T2 |
| 83 | 1+(−3.96−9.56i)T+(−58.6+58.6i)T2 |
| 89 | 1+(−2.57−2.57i)T+89iT2 |
| 97 | 1+4.78T+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
−11.32839242029512589671728080924, −10.65115824466677835446296839801, −9.983271835239808889002705531672, −8.457197005507111537124567492106, −8.095477695592259499108866134250, −7.16622256602311584914925553112, −6.38306658998642634088034468107, −5.21676614227267917731834353907, −3.67779506186302128395467432611, −2.54459557777677910376080298925,
0.29847046513722889623685218790, 2.28488471449049277310502117920, 3.45510735130441537155695621354, 4.59090305386043785297689449930, 5.45179514737022286917168761577, 7.49976564861194469419294701260, 8.395470082705262008465803432532, 8.791590954287980799698199829369, 9.814633187374381444352115199929, 10.49370001985986048820024276384