L(s) = 1 | + (−4 + 6.92i)2-s + (−13.5 − 23.3i)3-s + (−31.9 − 55.4i)4-s + (−235 + 407. i)5-s + 216·6-s + 511.·8-s + (−364.5 + 631. i)9-s + (−1.87e3 − 3.25e3i)10-s + (3.63e3 + 6.29e3i)11-s + (−864. + 1.49e3i)12-s + 1.13e4·13-s + 1.26e4·15-s + (−2.04e3 + 3.54e3i)16-s + (−1.04e4 − 1.80e4i)17-s + (−2.91e3 − 5.05e3i)18-s + (6.84e3 − 1.18e4i)19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.840 + 1.45i)5-s + 0.408·6-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.594 − 1.02i)10-s + (0.823 + 1.42i)11-s + (−0.144 + 0.249i)12-s + 1.43·13-s + 0.970·15-s + (−0.125 + 0.216i)16-s + (−0.514 − 0.891i)17-s + (−0.117 − 0.204i)18-s + (0.228 − 0.396i)19-s + ⋯ |
Λ(s)=(=(294s/2ΓC(s)L(s)(0.605−0.795i)Λ(8−s)
Λ(s)=(=(294s/2ΓC(s+7/2)L(s)(0.605−0.795i)Λ(1−s)
Degree: |
2 |
Conductor: |
294
= 2⋅3⋅72
|
Sign: |
0.605−0.795i
|
Analytic conductor: |
91.8411 |
Root analytic conductor: |
9.58338 |
Motivic weight: |
7 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ294(79,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 294, ( :7/2), 0.605−0.795i)
|
Particular Values
L(4) |
≈ |
1.364934233 |
L(21) |
≈ |
1.364934233 |
L(29) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(4−6.92i)T |
| 3 | 1+(13.5+23.3i)T |
| 7 | 1 |
good | 5 | 1+(235−407.i)T+(−3.90e4−6.76e4i)T2 |
| 11 | 1+(−3.63e3−6.29e3i)T+(−9.74e6+1.68e7i)T2 |
| 13 | 1−1.13e4T+6.27e7T2 |
| 17 | 1+(1.04e4+1.80e4i)T+(−2.05e8+3.55e8i)T2 |
| 19 | 1+(−6.84e3+1.18e4i)T+(−4.46e8−7.74e8i)T2 |
| 23 | 1+(−2.33e4+4.04e4i)T+(−1.70e9−2.94e9i)T2 |
| 29 | 1−2.77e3T+1.72e10T2 |
| 31 | 1+(−5.61e3−9.72e3i)T+(−1.37e10+2.38e10i)T2 |
| 37 | 1+(−1.19e5+2.07e5i)T+(−4.74e10−8.22e10i)T2 |
| 41 | 1−5.29e4T+1.94e11T2 |
| 43 | 1+8.74e5T+2.71e11T2 |
| 47 | 1+(−2.25e5+3.90e5i)T+(−2.53e11−4.38e11i)T2 |
| 53 | 1+(−9.71e5−1.68e6i)T+(−5.87e11+1.01e12i)T2 |
| 59 | 1+(7.81e5+1.35e6i)T+(−1.24e12+2.15e12i)T2 |
| 61 | 1+(−1.59e6+2.75e6i)T+(−1.57e12−2.72e12i)T2 |
| 67 | 1+(1.06e6+1.84e6i)T+(−3.03e12+5.24e12i)T2 |
| 71 | 1−5.69e6T+9.09e12T2 |
| 73 | 1+(−1.12e6−1.95e6i)T+(−5.52e12+9.56e12i)T2 |
| 79 | 1+(6.64e4−1.15e5i)T+(−9.60e12−1.66e13i)T2 |
| 83 | 1−6.95e6T+2.71e13T2 |
| 89 | 1+(−5.31e6+9.20e6i)T+(−2.21e13−3.83e13i)T2 |
| 97 | 1−2.40e6T+8.07e13T2 |
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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
−10.86006216706538622409188130178, −9.727562745139641598057234299112, −8.598333005048052201768003557655, −7.47500009446721796015196845769, −6.87135634342503276146475293179, −6.31425245324834374931841370402, −4.70136688283467577368310452884, −3.52466870744652224539634074109, −2.10390646114507674990508447111, −0.61270414574465112540363598177,
0.72262259556792063577564694109, 1.33770567363112869478181544509, 3.56318226180308977580435819603, 3.96039781209892456861945649901, 5.23744176279373040365720101496, 6.32474718318988183144196075391, 8.133622213881373387016241433677, 8.616944620973555812483758227212, 9.267063838157835864596982899938, 10.59142193972083044078893748685