L(s) = 1 | + (−0.956 − 1.65i)2-s + (−0.830 + 1.43i)4-s + (−1.54 + 1.61i)5-s + (1.11 − 2.39i)7-s − 0.650·8-s + (4.15 + 1.00i)10-s + (−2.79 − 1.61i)11-s − 4.86·13-s + (−5.04 + 0.439i)14-s + (2.28 + 3.95i)16-s + (−0.631 − 0.364i)17-s + (−6.81 + 3.93i)19-s + (−1.04 − 3.56i)20-s + 6.17i·22-s + (2.43 + 4.21i)23-s + ⋯ |
L(s) = 1 | + (−0.676 − 1.17i)2-s + (−0.415 + 0.718i)4-s + (−0.689 + 0.723i)5-s + (0.422 − 0.906i)7-s − 0.229·8-s + (1.31 + 0.318i)10-s + (−0.842 − 0.486i)11-s − 1.34·13-s + (−1.34 + 0.117i)14-s + (0.570 + 0.988i)16-s + (−0.153 − 0.0884i)17-s + (−1.56 + 0.902i)19-s + (−0.234 − 0.796i)20-s + 1.31i·22-s + (0.507 + 0.879i)23-s + ⋯ |
Λ(s)=(=(315s/2ΓC(s)L(s)(−0.557−0.830i)Λ(2−s)
Λ(s)=(=(315s/2ΓC(s+1/2)L(s)(−0.557−0.830i)Λ(1−s)
Degree: |
2 |
Conductor: |
315
= 32⋅5⋅7
|
Sign: |
−0.557−0.830i
|
Analytic conductor: |
2.51528 |
Root analytic conductor: |
1.58596 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ315(269,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 315, ( :1/2), −0.557−0.830i)
|
Particular Values
L(1) |
≈ |
0.0786014+0.147490i |
L(21) |
≈ |
0.0786014+0.147490i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 5 | 1+(1.54−1.61i)T |
| 7 | 1+(−1.11+2.39i)T |
good | 2 | 1+(0.956+1.65i)T+(−1+1.73i)T2 |
| 11 | 1+(2.79+1.61i)T+(5.5+9.52i)T2 |
| 13 | 1+4.86T+13T2 |
| 17 | 1+(0.631+0.364i)T+(8.5+14.7i)T2 |
| 19 | 1+(6.81−3.93i)T+(9.5−16.4i)T2 |
| 23 | 1+(−2.43−4.21i)T+(−11.5+19.9i)T2 |
| 29 | 1+7.75iT−29T2 |
| 31 | 1+(1.23+0.714i)T+(15.5+26.8i)T2 |
| 37 | 1+(−2.74+1.58i)T+(18.5−32.0i)T2 |
| 41 | 1+1.40T+41T2 |
| 43 | 1−6.42iT−43T2 |
| 47 | 1+(4.21−2.43i)T+(23.5−40.7i)T2 |
| 53 | 1+(0.760−1.31i)T+(−26.5−45.8i)T2 |
| 59 | 1+(−3.15+5.45i)T+(−29.5−51.0i)T2 |
| 61 | 1+(2.05−1.18i)T+(30.5−52.8i)T2 |
| 67 | 1+(9.63+5.56i)T+(33.5+58.0i)T2 |
| 71 | 1+10.1iT−71T2 |
| 73 | 1+(−6.91+11.9i)T+(−36.5−63.2i)T2 |
| 79 | 1+(1.99+3.44i)T+(−39.5+68.4i)T2 |
| 83 | 1+4.19iT−83T2 |
| 89 | 1+(−5.63−9.76i)T+(−44.5+77.0i)T2 |
| 97 | 1−2.21T+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
−10.91149138657764704108660378394, −10.38533100045486043805009776843, −9.549361580553133393060760734698, −8.129487552532815984334667232789, −7.61252535342014103729407528286, −6.24787316823115470402671551018, −4.53231417966453697689878627617, −3.35380211681026482900260750646, −2.17415096035057271236499659581, −0.13903963184807509749755232770,
2.54851810698838254725226899564, 4.69446053045142824261650806906, 5.33207084956978544622069200864, 6.77782996607988014252542531610, 7.55147223321698243265918763851, 8.585548792940588613143102876837, 8.872741860591462669297785783294, 10.13280474775578537724341388480, 11.37627336968835103070580492877, 12.46622799604611728943784618597