L(s) = 1 | + (0.562 − 2.09i)2-s + (−1.57 − 0.717i)3-s + (−2.35 − 1.35i)4-s + (−0.142 + 0.529i)5-s + (−2.39 + 2.90i)6-s + (−3.03 − 3.03i)7-s + (−1.10 + 1.10i)8-s + (1.97 + 2.26i)9-s + (1.03 + 0.595i)10-s + (3.61 + 0.969i)11-s + (2.73 + 3.83i)12-s + (2.50 − 2.59i)13-s + (−8.06 + 4.65i)14-s + (0.604 − 0.733i)15-s + (−1.02 − 1.77i)16-s + (0.784 + 1.35i)17-s + ⋯ |
L(s) = 1 | + (0.397 − 1.48i)2-s + (−0.910 − 0.414i)3-s + (−1.17 − 0.679i)4-s + (−0.0635 + 0.237i)5-s + (−0.976 + 1.18i)6-s + (−1.14 − 1.14i)7-s + (−0.389 + 0.389i)8-s + (0.656 + 0.754i)9-s + (0.326 + 0.188i)10-s + (1.09 + 0.292i)11-s + (0.789 + 1.10i)12-s + (0.694 − 0.719i)13-s + (−2.15 + 1.24i)14-s + (0.155 − 0.189i)15-s + (−0.256 − 0.443i)16-s + (0.190 + 0.329i)17-s + ⋯ |
Λ(s)=(=(117s/2ΓC(s)L(s)(−0.966+0.256i)Λ(2−s)
Λ(s)=(=(117s/2ΓC(s+1/2)L(s)(−0.966+0.256i)Λ(1−s)
Degree: |
2 |
Conductor: |
117
= 32⋅13
|
Sign: |
−0.966+0.256i
|
Analytic conductor: |
0.934249 |
Root analytic conductor: |
0.966565 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ117(110,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 117, ( :1/2), −0.966+0.256i)
|
Particular Values
L(1) |
≈ |
0.114977−0.883211i |
L(21) |
≈ |
0.114977−0.883211i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(1.57+0.717i)T |
| 13 | 1+(−2.50+2.59i)T |
good | 2 | 1+(−0.562+2.09i)T+(−1.73−i)T2 |
| 5 | 1+(0.142−0.529i)T+(−4.33−2.5i)T2 |
| 7 | 1+(3.03+3.03i)T+7iT2 |
| 11 | 1+(−3.61−0.969i)T+(9.52+5.5i)T2 |
| 17 | 1+(−0.784−1.35i)T+(−8.5+14.7i)T2 |
| 19 | 1+(3.27+0.876i)T+(16.4+9.5i)T2 |
| 23 | 1−5.03T+23T2 |
| 29 | 1+(1.02−0.593i)T+(14.5−25.1i)T2 |
| 31 | 1+(1.47+0.395i)T+(26.8+15.5i)T2 |
| 37 | 1+(−1.31+0.351i)T+(32.0−18.5i)T2 |
| 41 | 1+(−3.76−3.76i)T+41iT2 |
| 43 | 1−10.3iT−43T2 |
| 47 | 1+(1.82+6.81i)T+(−40.7+23.5i)T2 |
| 53 | 1−5.04iT−53T2 |
| 59 | 1+(−2.80−10.4i)T+(−51.0+29.5i)T2 |
| 61 | 1+5.93T+61T2 |
| 67 | 1+(−2.25+2.25i)T−67iT2 |
| 71 | 1+(0.774−2.89i)T+(−61.4−35.5i)T2 |
| 73 | 1+(9.10+9.10i)T+73iT2 |
| 79 | 1+(−7.18+12.4i)T+(−39.5−68.4i)T2 |
| 83 | 1+(−13.7+3.67i)T+(71.8−41.5i)T2 |
| 89 | 1+(1.63+6.09i)T+(−77.0+44.5i)T2 |
| 97 | 1+(−2.31+2.31i)T−97iT2 |
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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
−13.12825991590670402008838635017, −12.02954403287060479236763634170, −10.90743180601369050631020579634, −10.51217430166102823218567074010, −9.360279443653027490533213843384, −7.24443509617392358682893600792, −6.27753700507350748606955153923, −4.46039529304288451619343806426, −3.31425686627339240584302105395, −1.09518927662725221292613882711,
3.91251047114714827228617894651, 5.28163574451157767498164207305, 6.28955747290940900241178799664, 6.77845538728736641771687700529, 8.736105246396045445599896586751, 9.346379296865148734988878386839, 11.03011209879660746813479866075, 12.19299437751694055051367182074, 13.01152091295700962877733925484, 14.33141301243837676362313387576