L(s) = 1 | + (0.866 − 0.5i)2-s + (0.523 + 1.65i)3-s + (0.499 − 0.866i)4-s + (−0.419 − 0.242i)5-s + (1.27 + 1.16i)6-s + (4.37 − 2.52i)7-s − 0.999i·8-s + (−2.45 + 1.72i)9-s − 0.484·10-s + (−2.78 + 1.60i)11-s + (1.69 + 0.372i)12-s + (−0.722 + 3.53i)13-s + (2.52 − 4.37i)14-s + (0.180 − 0.819i)15-s + (−0.5 − 0.866i)16-s + 4.20·17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.302 + 0.953i)3-s + (0.249 − 0.433i)4-s + (−0.187 − 0.108i)5-s + (0.521 + 0.476i)6-s + (1.65 − 0.955i)7-s − 0.353i·8-s + (−0.817 + 0.575i)9-s − 0.153·10-s + (−0.838 + 0.484i)11-s + (0.488 + 0.107i)12-s + (−0.200 + 0.979i)13-s + (0.675 − 1.16i)14-s + (0.0465 − 0.211i)15-s + (−0.125 − 0.216i)16-s + 1.01·17-s + ⋯ |
Λ(s)=(=(234s/2ΓC(s)L(s)(0.998−0.0628i)Λ(2−s)
Λ(s)=(=(234s/2ΓC(s+1/2)L(s)(0.998−0.0628i)Λ(1−s)
Degree: |
2 |
Conductor: |
234
= 2⋅32⋅13
|
Sign: |
0.998−0.0628i
|
Analytic conductor: |
1.86849 |
Root analytic conductor: |
1.36693 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ234(103,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 234, ( :1/2), 0.998−0.0628i)
|
Particular Values
L(1) |
≈ |
1.92307+0.0604946i |
L(21) |
≈ |
1.92307+0.0604946i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.866+0.5i)T |
| 3 | 1+(−0.523−1.65i)T |
| 13 | 1+(0.722−3.53i)T |
good | 5 | 1+(0.419+0.242i)T+(2.5+4.33i)T2 |
| 7 | 1+(−4.37+2.52i)T+(3.5−6.06i)T2 |
| 11 | 1+(2.78−1.60i)T+(5.5−9.52i)T2 |
| 17 | 1−4.20T+17T2 |
| 19 | 1+3.21iT−19T2 |
| 23 | 1+(3.13−5.43i)T+(−11.5−19.9i)T2 |
| 29 | 1+(2.29+3.97i)T+(−14.5+25.1i)T2 |
| 31 | 1+(5.61+3.24i)T+(15.5+26.8i)T2 |
| 37 | 1−2.08iT−37T2 |
| 41 | 1+(9.57+5.52i)T+(20.5+35.5i)T2 |
| 43 | 1+(−4.73−8.19i)T+(−21.5+37.2i)T2 |
| 47 | 1+(4.57−2.64i)T+(23.5−40.7i)T2 |
| 53 | 1−6.41T+53T2 |
| 59 | 1+(−3.13−1.81i)T+(29.5+51.0i)T2 |
| 61 | 1+(−0.500−0.867i)T+(−30.5+52.8i)T2 |
| 67 | 1+(−0.936−0.540i)T+(33.5+58.0i)T2 |
| 71 | 1+4.63iT−71T2 |
| 73 | 1−0.325iT−73T2 |
| 79 | 1+(−3.91−6.78i)T+(−39.5+68.4i)T2 |
| 83 | 1+(5.08−2.93i)T+(41.5−71.8i)T2 |
| 89 | 1−8.42iT−89T2 |
| 97 | 1+(−11.3+6.52i)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
−11.83908769594363061288719182794, −11.30445126111200316527822951344, −10.38788890916419891068670479641, −9.582225356780364229103601943021, −8.139444100513985447808909912517, −7.41795674966070994729076327883, −5.45805951283634357084208518174, −4.62317197358608319688315251680, −3.84797188711701029667793640264, −2.05011077504871460122076348568,
1.95972902279580151126550800761, 3.30630974924463602944335801523, 5.26295149731319636375822914671, 5.72883437381789538563360127431, 7.38140907671964469391248692635, 8.065947964994334408109876822105, 8.637791747780199068349178443578, 10.53446899025008018972216725863, 11.58509894114505109941267893331, 12.27444030076020764473218228433