L(s) = 1 | + (1 + 1.73i)3-s + 3·5-s + (−2 + 3.46i)7-s + (−0.499 + 0.866i)9-s + (3 + 5.19i)15-s + (−1.5 + 2.59i)17-s + (1 − 1.73i)19-s − 7.99·21-s + (3 + 5.19i)23-s + 4·25-s + 4.00·27-s + (−4.5 − 7.79i)29-s − 2·31-s + (−6 + 10.3i)35-s + (−3.5 − 6.06i)37-s + ⋯ |
L(s) = 1 | + (0.577 + 0.999i)3-s + 1.34·5-s + (−0.755 + 1.30i)7-s + (−0.166 + 0.288i)9-s + (0.774 + 1.34i)15-s + (−0.363 + 0.630i)17-s + (0.229 − 0.397i)19-s − 1.74·21-s + (0.625 + 1.08i)23-s + 0.800·25-s + 0.769·27-s + (−0.835 − 1.44i)29-s − 0.359·31-s + (−1.01 + 1.75i)35-s + (−0.575 − 0.996i)37-s + ⋯ |
Λ(s)=(=(676s/2ΓC(s)L(s)(0.0128−0.999i)Λ(2−s)
Λ(s)=(=(676s/2ΓC(s+1/2)L(s)(0.0128−0.999i)Λ(1−s)
Degree: |
2 |
Conductor: |
676
= 22⋅132
|
Sign: |
0.0128−0.999i
|
Analytic conductor: |
5.39788 |
Root analytic conductor: |
2.32333 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ676(653,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 676, ( :1/2), 0.0128−0.999i)
|
Particular Values
L(1) |
≈ |
1.47685+1.45804i |
L(21) |
≈ |
1.47685+1.45804i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 13 | 1 |
good | 3 | 1+(−1−1.73i)T+(−1.5+2.59i)T2 |
| 5 | 1−3T+5T2 |
| 7 | 1+(2−3.46i)T+(−3.5−6.06i)T2 |
| 11 | 1+(−5.5+9.52i)T2 |
| 17 | 1+(1.5−2.59i)T+(−8.5−14.7i)T2 |
| 19 | 1+(−1+1.73i)T+(−9.5−16.4i)T2 |
| 23 | 1+(−3−5.19i)T+(−11.5+19.9i)T2 |
| 29 | 1+(4.5+7.79i)T+(−14.5+25.1i)T2 |
| 31 | 1+2T+31T2 |
| 37 | 1+(3.5+6.06i)T+(−18.5+32.0i)T2 |
| 41 | 1+(−1.5−2.59i)T+(−20.5+35.5i)T2 |
| 43 | 1+(−2+3.46i)T+(−21.5−37.2i)T2 |
| 47 | 1−6T+47T2 |
| 53 | 1−9T+53T2 |
| 59 | 1+(−29.5−51.0i)T2 |
| 61 | 1+(2.5−4.33i)T+(−30.5−52.8i)T2 |
| 67 | 1+(−1−1.73i)T+(−33.5+58.0i)T2 |
| 71 | 1+(3−5.19i)T+(−35.5−61.4i)T2 |
| 73 | 1−T+73T2 |
| 79 | 1+4T+79T2 |
| 83 | 1+12T+83T2 |
| 89 | 1+(−3−5.19i)T+(−44.5+77.0i)T2 |
| 97 | 1+(−7+12.1i)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
−10.35560157465230223194996454498, −9.630453088540719006615741088672, −9.209257756324542347096255573778, −8.631328607182920934063792943016, −7.09528233438551949372320990265, −5.86975554491085492382673460408, −5.53909783371407478066925571533, −4.11263979243731739002034386271, −2.97597831775266201140342160886, −2.07711158243366803850872861099,
1.10678632525517700864777588364, 2.26959389558707272582976937652, 3.36708266486775525465195576908, 4.82725718464144349283243456518, 6.05537491102806553302358037419, 6.96112979680475448652084198151, 7.32374642384608213755406093035, 8.612550988928739262052648196416, 9.406321567730820213202728551975, 10.27101224286537867897798549883