L(s) = 1 | + (−1.22 + 0.707i)2-s + (−2.66 − 1.37i)3-s + (0.999 − 1.73i)4-s − 2.72i·5-s + (4.23 − 0.207i)6-s + (−4.37 + 5.46i)7-s + 2.82i·8-s + (5.24 + 7.31i)9-s + (1.92 + 3.34i)10-s − 2.91i·11-s + (−5.04 + 3.25i)12-s + (10.4 + 18.1i)13-s + (1.49 − 9.78i)14-s + (−3.73 + 7.27i)15-s + (−2.00 − 3.46i)16-s + (−24.7 + 14.2i)17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.889 − 0.457i)3-s + (0.249 − 0.433i)4-s − 0.545i·5-s + (0.706 − 0.0346i)6-s + (−0.625 + 0.780i)7-s + 0.353i·8-s + (0.582 + 0.813i)9-s + (0.192 + 0.334i)10-s − 0.264i·11-s + (−0.420 + 0.270i)12-s + (0.805 + 1.39i)13-s + (0.106 − 0.698i)14-s + (−0.249 + 0.485i)15-s + (−0.125 − 0.216i)16-s + (−1.45 + 0.841i)17-s + ⋯ |
Λ(s)=(=(126s/2ΓC(s)L(s)(−0.350−0.936i)Λ(3−s)
Λ(s)=(=(126s/2ΓC(s+1)L(s)(−0.350−0.936i)Λ(1−s)
Degree: |
2 |
Conductor: |
126
= 2⋅32⋅7
|
Sign: |
−0.350−0.936i
|
Analytic conductor: |
3.43325 |
Root analytic conductor: |
1.85290 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ126(65,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 126, ( :1), −0.350−0.936i)
|
Particular Values
L(23) |
≈ |
0.255685+0.368864i |
L(21) |
≈ |
0.255685+0.368864i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(1.22−0.707i)T |
| 3 | 1+(2.66+1.37i)T |
| 7 | 1+(4.37−5.46i)T |
good | 5 | 1+2.72iT−25T2 |
| 11 | 1+2.91iT−121T2 |
| 13 | 1+(−10.4−18.1i)T+(−84.5+146.i)T2 |
| 17 | 1+(24.7−14.2i)T+(144.5−250.i)T2 |
| 19 | 1+(17.7−30.7i)T+(−180.5−312.i)T2 |
| 23 | 1+15.7iT−529T2 |
| 29 | 1+(−18.1−10.4i)T+(420.5+728.i)T2 |
| 31 | 1+(−6.23+10.7i)T+(−480.5−832.i)T2 |
| 37 | 1+(5.80−10.0i)T+(−684.5−1.18e3i)T2 |
| 41 | 1+(−26.4+15.2i)T+(840.5−1.45e3i)T2 |
| 43 | 1+(12.6−21.9i)T+(−924.5−1.60e3i)T2 |
| 47 | 1+(73.2−42.3i)T+(1.10e3−1.91e3i)T2 |
| 53 | 1+(15.1−8.77i)T+(1.40e3−2.43e3i)T2 |
| 59 | 1+(37.3+21.5i)T+(1.74e3+3.01e3i)T2 |
| 61 | 1+(15.1+26.2i)T+(−1.86e3+3.22e3i)T2 |
| 67 | 1+(43.2−74.8i)T+(−2.24e3−3.88e3i)T2 |
| 71 | 1−1.24iT−5.04e3T2 |
| 73 | 1+(6.48+11.2i)T+(−2.66e3+4.61e3i)T2 |
| 79 | 1+(51.7+89.6i)T+(−3.12e3+5.40e3i)T2 |
| 83 | 1+(−35.2−20.3i)T+(3.44e3+5.96e3i)T2 |
| 89 | 1+(15.5+8.95i)T+(3.96e3+6.85e3i)T2 |
| 97 | 1+(2.62−4.54i)T+(−4.70e3−8.14e3i)T2 |
show more | |
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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
−13.16095759505996875924223154953, −12.41327234136455438376213761769, −11.36021554330148016337625082590, −10.39773729353170624217941633582, −9.015334669632577462091068983262, −8.284243616743553992314375296061, −6.47907150439339588496337902904, −6.18332500186691394500011137479, −4.50368887409262620278906510233, −1.76180490719361802161953471175,
0.41781446947808200925621359262, 3.11357484311845391878798702782, 4.62758213349960304573528136992, 6.40414092329974976015936988398, 7.14930280461051522701248491702, 8.831854643488442595383620072135, 9.974157058281824167638447140290, 10.80036882524349645733547713326, 11.29369109880600017997612665532, 12.77097819408558649737459594927