L(s) = 1 | + (−0.313 − 0.542i)2-s + (0.803 − 1.39i)4-s + 5-s + (−2.21 + 3.83i)7-s − 2.26·8-s + (−0.313 − 0.542i)10-s + (−3.02 − 5.24i)11-s + (2.27 − 2.80i)13-s + 2.77·14-s + (−0.898 − 1.55i)16-s + (2.92 − 5.05i)17-s + (1.80 − 3.12i)19-s + (0.803 − 1.39i)20-s + (−1.89 + 3.28i)22-s + (−1.13 − 1.95i)23-s + ⋯ |
L(s) = 1 | + (−0.221 − 0.383i)2-s + (0.401 − 0.695i)4-s + 0.447·5-s + (−0.837 + 1.45i)7-s − 0.799·8-s + (−0.0991 − 0.171i)10-s + (−0.913 − 1.58i)11-s + (0.629 − 0.776i)13-s + 0.742·14-s + (−0.224 − 0.389i)16-s + (0.708 − 1.22i)17-s + (0.413 − 0.716i)19-s + (0.179 − 0.311i)20-s + (−0.404 + 0.701i)22-s + (−0.235 − 0.408i)23-s + ⋯ |
Λ(s)=(=(585s/2ΓC(s)L(s)(−0.401+0.915i)Λ(2−s)
Λ(s)=(=(585s/2ΓC(s+1/2)L(s)(−0.401+0.915i)Λ(1−s)
Degree: |
2 |
Conductor: |
585
= 32⋅5⋅13
|
Sign: |
−0.401+0.915i
|
Analytic conductor: |
4.67124 |
Root analytic conductor: |
2.16130 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ585(406,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 585, ( :1/2), −0.401+0.915i)
|
Particular Values
L(1) |
≈ |
0.646413−0.989093i |
L(21) |
≈ |
0.646413−0.989093i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 5 | 1−T |
| 13 | 1+(−2.27+2.80i)T |
good | 2 | 1+(0.313+0.542i)T+(−1+1.73i)T2 |
| 7 | 1+(2.21−3.83i)T+(−3.5−6.06i)T2 |
| 11 | 1+(3.02+5.24i)T+(−5.5+9.52i)T2 |
| 17 | 1+(−2.92+5.05i)T+(−8.5−14.7i)T2 |
| 19 | 1+(−1.80+3.12i)T+(−9.5−16.4i)T2 |
| 23 | 1+(1.13+1.95i)T+(−11.5+19.9i)T2 |
| 29 | 1+(4.04+7.00i)T+(−14.5+25.1i)T2 |
| 31 | 1−6.45T+31T2 |
| 37 | 1+(−0.898−1.55i)T+(−18.5+32.0i)T2 |
| 41 | 1+(−3.99−6.92i)T+(−20.5+35.5i)T2 |
| 43 | 1+(3.24−5.61i)T+(−21.5−37.2i)T2 |
| 47 | 1+3.22T+47T2 |
| 53 | 1−10.0T+53T2 |
| 59 | 1+(1.56−2.70i)T+(−29.5−51.0i)T2 |
| 61 | 1+(−1.22+2.12i)T+(−30.5−52.8i)T2 |
| 67 | 1+(−2.16−3.75i)T+(−33.5+58.0i)T2 |
| 71 | 1+(4.24−7.35i)T+(−35.5−61.4i)T2 |
| 73 | 1+0.819T+73T2 |
| 79 | 1+8.42T+79T2 |
| 83 | 1−2.35T+83T2 |
| 89 | 1+(0.386+0.669i)T+(−44.5+77.0i)T2 |
| 97 | 1+(−3.16+5.47i)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
−10.29320294575057747898372949032, −9.667747676795199349613147421695, −8.888005058639855975631408821325, −7.998141459004318348391460688191, −6.43595641229417738048622737914, −5.80445896143313874752251422607, −5.29528570803006147693409112089, −2.91354768050988957628691420427, −2.73814304254478478460589027174, −0.69843594285636750727066279333,
1.81544647822290738232505139816, 3.38912327447566268442115279191, 4.18099297643239660347377868047, 5.71821933856602873730027077643, 6.78314005838590830084075938010, 7.30081992354382502589821205358, 8.095528496786361692812514600842, 9.294588216893771859651578108735, 10.17846407797431878963646824252, 10.63748168655951209722069251234