L(s) = 1 | + (1.63 + 1.18i)3-s + (−1.62 − 0.527i)5-s + (3.70 − 2.68i)7-s + (0.333 + 1.02i)9-s + (2.77 + 1.81i)11-s + (−0.147 − 0.455i)13-s + (−2.02 − 2.79i)15-s + (2.14 + 0.696i)17-s + (−1.24 + 1.70i)19-s + 9.24·21-s + 7.45i·23-s + (−1.68 − 1.22i)25-s + (1.19 − 3.68i)27-s + (−3.30 + 2.40i)29-s + (−3.84 + 1.24i)31-s + ⋯ |
L(s) = 1 | + (0.943 + 0.685i)3-s + (−0.726 − 0.236i)5-s + (1.39 − 1.01i)7-s + (0.111 + 0.342i)9-s + (0.836 + 0.547i)11-s + (−0.0410 − 0.126i)13-s + (−0.523 − 0.720i)15-s + (0.519 + 0.168i)17-s + (−0.284 + 0.391i)19-s + 2.01·21-s + 1.55i·23-s + (−0.337 − 0.244i)25-s + (0.230 − 0.709i)27-s + (−0.614 + 0.446i)29-s + (−0.690 + 0.224i)31-s + ⋯ |
Λ(s)=(=(352s/2ΓC(s)L(s)(0.976−0.215i)Λ(2−s)
Λ(s)=(=(352s/2ΓC(s+1/2)L(s)(0.976−0.215i)Λ(1−s)
Degree: |
2 |
Conductor: |
352
= 25⋅11
|
Sign: |
0.976−0.215i
|
Analytic conductor: |
2.81073 |
Root analytic conductor: |
1.67652 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ352(303,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 352, ( :1/2), 0.976−0.215i)
|
Particular Values
L(1) |
≈ |
1.82442+0.198929i |
L(21) |
≈ |
1.82442+0.198929i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 11 | 1+(−2.77−1.81i)T |
good | 3 | 1+(−1.63−1.18i)T+(0.927+2.85i)T2 |
| 5 | 1+(1.62+0.527i)T+(4.04+2.93i)T2 |
| 7 | 1+(−3.70+2.68i)T+(2.16−6.65i)T2 |
| 13 | 1+(0.147+0.455i)T+(−10.5+7.64i)T2 |
| 17 | 1+(−2.14−0.696i)T+(13.7+9.99i)T2 |
| 19 | 1+(1.24−1.70i)T+(−5.87−18.0i)T2 |
| 23 | 1−7.45iT−23T2 |
| 29 | 1+(3.30−2.40i)T+(8.96−27.5i)T2 |
| 31 | 1+(3.84−1.24i)T+(25.0−18.2i)T2 |
| 37 | 1+(3.89+5.36i)T+(−11.4+35.1i)T2 |
| 41 | 1+(3.72−5.12i)T+(−12.6−38.9i)T2 |
| 43 | 1+5.32iT−43T2 |
| 47 | 1+(−2.59+3.56i)T+(−14.5−44.6i)T2 |
| 53 | 1+(−1.67+0.545i)T+(42.8−31.1i)T2 |
| 59 | 1+(−6.30+4.58i)T+(18.2−56.1i)T2 |
| 61 | 1+(0.782−2.40i)T+(−49.3−35.8i)T2 |
| 67 | 1+4.41T+67T2 |
| 71 | 1+(4.63+1.50i)T+(57.4+41.7i)T2 |
| 73 | 1+(4.11+5.65i)T+(−22.5+69.4i)T2 |
| 79 | 1+(−5.15−15.8i)T+(−63.9+46.4i)T2 |
| 83 | 1+(5.14+1.67i)T+(67.1+48.7i)T2 |
| 89 | 1+13.0T+89T2 |
| 97 | 1+(−1.68−5.17i)T+(−78.4+57.0i)T2 |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−11.48687914507304626772732695050, −10.52420882743968963736655367738, −9.610851740068528235475136925062, −8.645600257669039453801862241284, −7.86687989928302382979864525869, −7.15323444985431647796746314407, −5.29333731524742810817559270573, −4.07508808628458179286058896073, −3.71040767540842620908771391603, −1.62680174736185235223632349853,
1.72130784877759638776693313205, 2.88587446305526921560828621295, 4.27726343701190746222562498586, 5.57860083968228509986276774725, 6.94064304227840441442972793755, 7.910515292097411973280577716157, 8.480759140006065559365978087151, 9.150682636038027336167383531497, 10.78565182321918911897729166968, 11.61212575290281280667854334063