L(s) = 1 | + (−0.417 − 1.35i)2-s + (−1.00 − 0.580i)3-s + (−1.65 + 1.12i)4-s + 5-s + (−0.364 + 1.60i)6-s + (−2.74 + 1.58i)7-s + (2.21 + 1.76i)8-s + (−0.826 − 1.43i)9-s + (−0.417 − 1.35i)10-s + (−1.23 + 2.13i)11-s + (2.31 − 0.175i)12-s + (3.60 + 0.150i)13-s + (3.28 + 3.04i)14-s + (−1.00 − 0.580i)15-s + (1.45 − 3.72i)16-s + (0.369 + 0.639i)17-s + ⋯ |
L(s) = 1 | + (−0.295 − 0.955i)2-s + (−0.580 − 0.335i)3-s + (−0.825 + 0.564i)4-s + 0.447·5-s + (−0.148 + 0.653i)6-s + (−1.03 + 0.598i)7-s + (0.782 + 0.622i)8-s + (−0.275 − 0.477i)9-s + (−0.132 − 0.427i)10-s + (−0.371 + 0.643i)11-s + (0.668 − 0.0506i)12-s + (0.999 + 0.0416i)13-s + (0.877 + 0.813i)14-s + (−0.259 − 0.149i)15-s + (0.363 − 0.931i)16-s + (0.0895 + 0.155i)17-s + ⋯ |
Λ(s)=(=(520s/2ΓC(s)L(s)(0.984+0.176i)Λ(2−s)
Λ(s)=(=(520s/2ΓC(s+1/2)L(s)(0.984+0.176i)Λ(1−s)
Degree: |
2 |
Conductor: |
520
= 23⋅5⋅13
|
Sign: |
0.984+0.176i
|
Analytic conductor: |
4.15222 |
Root analytic conductor: |
2.03769 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ520(381,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 520, ( :1/2), 0.984+0.176i)
|
Particular Values
L(1) |
≈ |
0.789850−0.0700978i |
L(21) |
≈ |
0.789850−0.0700978i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.417+1.35i)T |
| 5 | 1−T |
| 13 | 1+(−3.60−0.150i)T |
good | 3 | 1+(1.00+0.580i)T+(1.5+2.59i)T2 |
| 7 | 1+(2.74−1.58i)T+(3.5−6.06i)T2 |
| 11 | 1+(1.23−2.13i)T+(−5.5−9.52i)T2 |
| 17 | 1+(−0.369−0.639i)T+(−8.5+14.7i)T2 |
| 19 | 1+(−4.31−7.47i)T+(−9.5+16.4i)T2 |
| 23 | 1+(−2.88+4.99i)T+(−11.5−19.9i)T2 |
| 29 | 1+(−1.59−0.919i)T+(14.5+25.1i)T2 |
| 31 | 1−9.05iT−31T2 |
| 37 | 1+(−1.35+2.35i)T+(−18.5−32.0i)T2 |
| 41 | 1+(−0.165−0.0952i)T+(20.5+35.5i)T2 |
| 43 | 1+(−7.43+4.29i)T+(21.5−37.2i)T2 |
| 47 | 1+8.23iT−47T2 |
| 53 | 1−4.18iT−53T2 |
| 59 | 1+(−5.72−9.91i)T+(−29.5+51.0i)T2 |
| 61 | 1+(4.03−2.32i)T+(30.5−52.8i)T2 |
| 67 | 1+(1.66−2.88i)T+(−33.5−58.0i)T2 |
| 71 | 1+(3.24−1.87i)T+(35.5−61.4i)T2 |
| 73 | 1−1.03iT−73T2 |
| 79 | 1+9.18T+79T2 |
| 83 | 1−1.79T+83T2 |
| 89 | 1+(−12.1−7.04i)T+(44.5+77.0i)T2 |
| 97 | 1+(14.8−8.56i)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
−10.70408593149410961255691857440, −10.14719640343093552784391773900, −9.186102311886725946143713394255, −8.554145540660360025307055018713, −7.18072753894081425954899925087, −6.10278547030343444839680252753, −5.33789166170732443574379479574, −3.75501491186510882235311734467, −2.77182955663918343285431579113, −1.23923588230200569902590736599,
0.65891489725841402378564813176, 3.13805770120695634449426061513, 4.54511139913823067298989013877, 5.57855847370836146455397900055, 6.18277077057870354647374849099, 7.16777087203406723294038441384, 8.117923761009004618101914412745, 9.294603444756853887049708910927, 9.763780151243386260499590309574, 10.86436116292131413821118348168