L(s) = 1 | + (0.289 − 0.502i)2-s + (0.946 − 1.63i)3-s + (0.831 + 1.44i)4-s + 1.47·5-s + (−0.548 − 0.950i)6-s + 2.12·8-s + (−0.289 − 0.502i)9-s + (0.427 − 0.739i)10-s + (−0.289 + 0.502i)11-s + 3.14·12-s + (−0.128 − 3.60i)13-s + (1.39 − 2.41i)15-s + (−1.04 + 1.81i)16-s + (0.598 + 1.03i)17-s − 0.336·18-s + (0.230 + 0.399i)19-s + ⋯ |
L(s) = 1 | + (0.204 − 0.355i)2-s + (0.546 − 0.946i)3-s + (0.415 + 0.720i)4-s + 0.659·5-s + (−0.223 − 0.387i)6-s + 0.751·8-s + (−0.0966 − 0.167i)9-s + (0.135 − 0.233i)10-s + (−0.0874 + 0.151i)11-s + 0.908·12-s + (−0.0357 − 0.999i)13-s + (0.359 − 0.623i)15-s + (−0.261 + 0.453i)16-s + (0.145 + 0.251i)17-s − 0.0792·18-s + (0.0528 + 0.0915i)19-s + ⋯ |
Λ(s)=(=(637s/2ΓC(s)L(s)(0.727+0.686i)Λ(2−s)
Λ(s)=(=(637s/2ΓC(s+1/2)L(s)(0.727+0.686i)Λ(1−s)
Degree: |
2 |
Conductor: |
637
= 72⋅13
|
Sign: |
0.727+0.686i
|
Analytic conductor: |
5.08647 |
Root analytic conductor: |
2.25532 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ637(295,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 637, ( :1/2), 0.727+0.686i)
|
Particular Values
L(1) |
≈ |
2.31562−0.919598i |
L(21) |
≈ |
2.31562−0.919598i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 7 | 1 |
| 13 | 1+(0.128+3.60i)T |
good | 2 | 1+(−0.289+0.502i)T+(−1−1.73i)T2 |
| 3 | 1+(−0.946+1.63i)T+(−1.5−2.59i)T2 |
| 5 | 1−1.47T+5T2 |
| 11 | 1+(0.289−0.502i)T+(−5.5−9.52i)T2 |
| 17 | 1+(−0.598−1.03i)T+(−8.5+14.7i)T2 |
| 19 | 1+(−0.230−0.399i)T+(−9.5+16.4i)T2 |
| 23 | 1+(1.18−2.05i)T+(−11.5−19.9i)T2 |
| 29 | 1+(−3.44+5.96i)T+(−14.5−25.1i)T2 |
| 31 | 1−4.44T+31T2 |
| 37 | 1+(4.58−7.93i)T+(−18.5−32.0i)T2 |
| 41 | 1+(−2.00+3.47i)T+(−20.5−35.5i)T2 |
| 43 | 1+(4.02+6.97i)T+(−21.5+37.2i)T2 |
| 47 | 1+11.5T+47T2 |
| 53 | 1+9.39T+53T2 |
| 59 | 1+(0.120+0.208i)T+(−29.5+51.0i)T2 |
| 61 | 1+(3.86+6.69i)T+(−30.5+52.8i)T2 |
| 67 | 1+(−0.724+1.25i)T+(−33.5−58.0i)T2 |
| 71 | 1+(−6.25−10.8i)T+(−35.5+61.4i)T2 |
| 73 | 1+3.69T+73T2 |
| 79 | 1−16.0T+79T2 |
| 83 | 1+15.4T+83T2 |
| 89 | 1+(1.24−2.15i)T+(−44.5−77.0i)T2 |
| 97 | 1+(7.82+13.5i)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
−10.43483234266453222337866063878, −9.782022238773617990460042799171, −8.248361771669931834203090836789, −8.048024213274473977572146526860, −7.01649251790707416892268119677, −6.17255330241092875729253294727, −4.88202757789733824037848100683, −3.45227645664299585687775169474, −2.50572230772502125393385968961, −1.59058216955735166432475461521,
1.66158221927596276142838311504, 2.98727383168121233517606937412, 4.34880512575019808267548990958, 5.09382296465203305565283742638, 6.21804703500295942248998175848, 6.87152506967442415187739369277, 8.161876495958905586269346762800, 9.273182458155870853294894801469, 9.724018601394590073266889043170, 10.48269374612427527758620041950