L(s) = 1 | + (19.7 + 60.8i)2-s + (1.19e3 + 866. i)3-s + (−3.31e3 + 2.40e3i)4-s + (−1.62e4 + 4.99e4i)5-s + (−2.91e4 + 8.97e4i)6-s + (−2.03e5 + 1.47e5i)7-s + (−2.12e5 − 1.54e5i)8-s + (1.79e5 + 5.52e5i)9-s − 3.36e6·10-s + (−7.55e5 − 5.82e6i)11-s − 6.04e6·12-s + (2.64e6 + 8.13e6i)13-s + (−1.29e7 − 9.44e6i)14-s + (−6.26e7 + 4.55e7i)15-s + (5.18e6 − 1.59e7i)16-s + (1.00e7 − 3.09e7i)17-s + ⋯ |
L(s) = 1 | + (0.218 + 0.672i)2-s + (0.944 + 0.686i)3-s + (−0.404 + 0.293i)4-s + (−0.464 + 1.42i)5-s + (−0.255 + 0.785i)6-s + (−0.652 + 0.474i)7-s + (−0.286 − 0.207i)8-s + (0.112 + 0.346i)9-s − 1.06·10-s + (−0.128 − 0.991i)11-s − 0.584·12-s + (0.151 + 0.467i)13-s + (−0.461 − 0.335i)14-s + (−1.42 + 1.03i)15-s + (0.0772 − 0.237i)16-s + (0.101 − 0.311i)17-s + ⋯ |
Λ(s)=(=(22s/2ΓC(s)L(s)(−0.757+0.652i)Λ(14−s)
Λ(s)=(=(22s/2ΓC(s+13/2)L(s)(−0.757+0.652i)Λ(1−s)
Degree: |
2 |
Conductor: |
22
= 2⋅11
|
Sign: |
−0.757+0.652i
|
Analytic conductor: |
23.5908 |
Root analytic conductor: |
4.85703 |
Motivic weight: |
13 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ22(5,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 22, ( :13/2), −0.757+0.652i)
|
Particular Values
L(7) |
≈ |
1.654964908 |
L(21) |
≈ |
1.654964908 |
L(215) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−19.7−60.8i)T |
| 11 | 1+(7.55e5+5.82e6i)T |
good | 3 | 1+(−1.19e3−866.i)T+(4.92e5+1.51e6i)T2 |
| 5 | 1+(1.62e4−4.99e4i)T+(−9.87e8−7.17e8i)T2 |
| 7 | 1+(2.03e5−1.47e5i)T+(2.99e10−9.21e10i)T2 |
| 13 | 1+(−2.64e6−8.13e6i)T+(−2.45e14+1.78e14i)T2 |
| 17 | 1+(−1.00e7+3.09e7i)T+(−8.01e15−5.82e15i)T2 |
| 19 | 1+(−2.79e8−2.02e8i)T+(1.29e16+3.99e16i)T2 |
| 23 | 1+2.93e8T+5.04e17T2 |
| 29 | 1+(2.88e9−2.09e9i)T+(3.17e18−9.75e18i)T2 |
| 31 | 1+(3.26e8+1.00e9i)T+(−1.97e19+1.43e19i)T2 |
| 37 | 1+(4.20e9−3.05e9i)T+(7.52e19−2.31e20i)T2 |
| 41 | 1+(2.43e10+1.77e10i)T+(2.85e20+8.79e20i)T2 |
| 43 | 1+6.71e10T+1.71e21T2 |
| 47 | 1+(−7.64e10−5.55e10i)T+(1.68e21+5.19e21i)T2 |
| 53 | 1+(−6.78e10−2.08e11i)T+(−2.10e22+1.53e22i)T2 |
| 59 | 1+(−3.13e11+2.28e11i)T+(3.24e22−9.98e22i)T2 |
| 61 | 1+(1.36e11−4.19e11i)T+(−1.30e23−9.51e22i)T2 |
| 67 | 1+1.03e12T+5.48e23T2 |
| 71 | 1+(6.07e11−1.86e12i)T+(−9.42e23−6.84e23i)T2 |
| 73 | 1+(3.11e11−2.26e11i)T+(5.16e23−1.59e24i)T2 |
| 79 | 1+(3.04e11+9.35e11i)T+(−3.77e24+2.74e24i)T2 |
| 83 | 1+(−4.10e11+1.26e12i)T+(−7.17e24−5.21e24i)T2 |
| 89 | 1−1.62e12T+2.19e25T2 |
| 97 | 1+(−3.05e12−9.39e12i)T+(−5.44e25+3.95e25i)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
−15.52097837415761155434518093541, −14.53384187749266485407948645087, −13.77722816188209064843406591509, −11.74208247691684275832909140543, −10.10661213084280514386338267608, −8.831783889549821927702768901795, −7.40379213668336355860704921039, −5.95726197856096274725124118757, −3.64503569489932124833072782418, −3.00290695997858487919666422350,
0.46059581780771040970308573645, 1.77699264427219907689491702731, 3.45016100409143803961530645710, 4.98877106454144206780554108531, 7.41904104274137910275167980343, 8.656164356417693667584251109739, 9.863740609190886174892613424734, 11.90305197104951954541509812397, 13.01258840736713622298333070223, 13.51484468698647899040264426243