L(s) = 1 | + (0.134 + 0.232i)2-s + (0.571 + 0.989i)3-s + (0.964 − 1.66i)4-s − 2.56·5-s + (−0.153 + 0.265i)6-s + 1.05·8-s + (0.846 − 1.46i)9-s + (−0.343 − 0.594i)10-s + (−1.97 − 3.41i)11-s + 2.20·12-s + (−3.15 − 1.74i)13-s + (−1.46 − 2.53i)15-s + (−1.78 − 3.09i)16-s + (−0.392 + 0.679i)17-s + 0.454·18-s + (3.74 − 6.49i)19-s + ⋯ |
L(s) = 1 | + (0.0947 + 0.164i)2-s + (0.329 + 0.571i)3-s + (0.482 − 0.834i)4-s − 1.14·5-s + (−0.0625 + 0.108i)6-s + 0.372·8-s + (0.282 − 0.488i)9-s + (−0.108 − 0.188i)10-s + (−0.594 − 1.03i)11-s + 0.636·12-s + (−0.874 − 0.484i)13-s + (−0.378 − 0.654i)15-s + (−0.446 − 0.773i)16-s + (−0.0952 + 0.164i)17-s + 0.107·18-s + (0.859 − 1.48i)19-s + ⋯ |
Λ(s)=(=(637s/2ΓC(s)L(s)(0.281+0.959i)Λ(2−s)
Λ(s)=(=(637s/2ΓC(s+1/2)L(s)(0.281+0.959i)Λ(1−s)
Degree: |
2 |
Conductor: |
637
= 72⋅13
|
Sign: |
0.281+0.959i
|
Analytic conductor: |
5.08647 |
Root analytic conductor: |
2.25532 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ637(393,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 637, ( :1/2), 0.281+0.959i)
|
Particular Values
L(1) |
≈ |
1.07276−0.802886i |
L(21) |
≈ |
1.07276−0.802886i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 7 | 1 |
| 13 | 1+(3.15+1.74i)T |
good | 2 | 1+(−0.134−0.232i)T+(−1+1.73i)T2 |
| 3 | 1+(−0.571−0.989i)T+(−1.5+2.59i)T2 |
| 5 | 1+2.56T+5T2 |
| 11 | 1+(1.97+3.41i)T+(−5.5+9.52i)T2 |
| 17 | 1+(0.392−0.679i)T+(−8.5−14.7i)T2 |
| 19 | 1+(−3.74+6.49i)T+(−9.5−16.4i)T2 |
| 23 | 1+(−3.97−6.88i)T+(−11.5+19.9i)T2 |
| 29 | 1+(1.17+2.03i)T+(−14.5+25.1i)T2 |
| 31 | 1+2.55T+31T2 |
| 37 | 1+(3.37+5.85i)T+(−18.5+32.0i)T2 |
| 41 | 1+(−1.21−2.11i)T+(−20.5+35.5i)T2 |
| 43 | 1+(−1.12+1.94i)T+(−21.5−37.2i)T2 |
| 47 | 1−1.31T+47T2 |
| 53 | 1−9.27T+53T2 |
| 59 | 1+(−4.48+7.76i)T+(−29.5−51.0i)T2 |
| 61 | 1+(4.72−8.18i)T+(−30.5−52.8i)T2 |
| 67 | 1+(−0.676−1.17i)T+(−33.5+58.0i)T2 |
| 71 | 1+(6.15−10.6i)T+(−35.5−61.4i)T2 |
| 73 | 1−0.768T+73T2 |
| 79 | 1−6.19T+79T2 |
| 83 | 1+1.07T+83T2 |
| 89 | 1+(3.83+6.63i)T+(−44.5+77.0i)T2 |
| 97 | 1+(−1.18+2.05i)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
−10.43987396738645960755460201715, −9.550430109277602844138019933427, −8.790756744976507445952713060216, −7.52164330670314438368212668847, −7.10410100987095381520116331331, −5.67561051926466589681581648015, −4.92701377484401482720796342877, −3.72065583743998813947118504416, −2.77443250128593687561481278172, −0.67176075289715635544913054514,
1.91465519376068749623184461451, 2.93681885303542192752732179998, 4.12055322569979237837222866411, 4.96739541670003912933788044764, 6.78957812018491143009843591386, 7.45880398169572080421942994470, 7.79837898838485130080248198842, 8.709473688446654480274541762303, 10.06184411355804235076940243072, 10.85322422558220201522926752692