L(s) = 1 | + (0.328 + 0.568i)2-s + (0.102 − 0.177i)3-s + (0.784 − 1.35i)4-s + (0.679 + 1.17i)5-s + 0.134·6-s + 2.34·8-s + (1.47 + 2.56i)9-s + (−0.446 + 0.772i)10-s + (0.952 − 1.65i)11-s + (−0.160 − 0.278i)12-s + 13-s + 0.278·15-s + (−0.800 − 1.38i)16-s + (−1.78 + 3.08i)17-s + (−0.970 + 1.68i)18-s + (−0.492 − 0.853i)19-s + ⋯ |
L(s) = 1 | + (0.231 + 0.401i)2-s + (0.0591 − 0.102i)3-s + (0.392 − 0.679i)4-s + (0.304 + 0.526i)5-s + 0.0548·6-s + 0.828·8-s + (0.493 + 0.853i)9-s + (−0.141 + 0.244i)10-s + (0.287 − 0.497i)11-s + (−0.0463 − 0.0803i)12-s + 0.277·13-s + 0.0718·15-s + (−0.200 − 0.346i)16-s + (−0.432 + 0.749i)17-s + (−0.228 + 0.396i)18-s + (−0.113 − 0.195i)19-s + ⋯ |
Λ(s)=(=(637s/2ΓC(s)L(s)(0.947−0.318i)Λ(2−s)
Λ(s)=(=(637s/2ΓC(s+1/2)L(s)(0.947−0.318i)Λ(1−s)
Degree: |
2 |
Conductor: |
637
= 72⋅13
|
Sign: |
0.947−0.318i
|
Analytic conductor: |
5.08647 |
Root analytic conductor: |
2.25532 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ637(508,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 637, ( :1/2), 0.947−0.318i)
|
Particular Values
L(1) |
≈ |
2.12497+0.347530i |
L(21) |
≈ |
2.12497+0.347530i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 7 | 1 |
| 13 | 1−T |
good | 2 | 1+(−0.328−0.568i)T+(−1+1.73i)T2 |
| 3 | 1+(−0.102+0.177i)T+(−1.5−2.59i)T2 |
| 5 | 1+(−0.679−1.17i)T+(−2.5+4.33i)T2 |
| 11 | 1+(−0.952+1.65i)T+(−5.5−9.52i)T2 |
| 17 | 1+(1.78−3.08i)T+(−8.5−14.7i)T2 |
| 19 | 1+(0.492+0.853i)T+(−9.5+16.4i)T2 |
| 23 | 1+(0.848+1.46i)T+(−11.5+19.9i)T2 |
| 29 | 1−6.54T+29T2 |
| 31 | 1+(3.84−6.66i)T+(−15.5−26.8i)T2 |
| 37 | 1+(−1.01−1.74i)T+(−18.5+32.0i)T2 |
| 41 | 1−9.88T+41T2 |
| 43 | 1+3.16T+43T2 |
| 47 | 1+(3.88+6.72i)T+(−23.5+40.7i)T2 |
| 53 | 1+(0.177−0.306i)T+(−26.5−45.8i)T2 |
| 59 | 1+(1.08−1.87i)T+(−29.5−51.0i)T2 |
| 61 | 1+(6.10+10.5i)T+(−30.5+52.8i)T2 |
| 67 | 1+(5.65−9.79i)T+(−33.5−58.0i)T2 |
| 71 | 1+9.05T+71T2 |
| 73 | 1+(−3.56+6.18i)T+(−36.5−63.2i)T2 |
| 79 | 1+(−2.69−4.67i)T+(−39.5+68.4i)T2 |
| 83 | 1+2.03T+83T2 |
| 89 | 1+(−3.44−5.97i)T+(−44.5+77.0i)T2 |
| 97 | 1+14.6T+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
−10.62561640143010208439573870431, −10.04655207863986345936528155736, −8.790165690490013739313562936043, −7.86190457149334874986839373063, −6.77904200944070560392274593739, −6.32485802986391215105583671072, −5.25026103693776246341549554763, −4.26942098262410053137293070000, −2.68141291035781260462684965015, −1.50231218827177621438038059862,
1.41226558446146016876854709612, 2.76326822632018647138816443641, 3.94053028275811712425050080207, 4.67954413496777973260434484352, 6.08726997742657281769606206600, 7.02907148231159407476500726971, 7.84203088891088240467489245030, 9.015645321429733422460718217558, 9.558833375308876548933023740099, 10.64669519257224733813597042567