L(s) = 1 | − i·2-s + 3-s − 4-s + 4.04i·5-s − i·6-s − 0.692i·7-s + i·8-s + 9-s + 4.04·10-s + 4.85i·11-s − 12-s − 0.692·14-s + 4.04i·15-s + 16-s − 7.38·17-s − i·18-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.577·3-s − 0.5·4-s + 1.81i·5-s − 0.408i·6-s − 0.261i·7-s + 0.353i·8-s + 0.333·9-s + 1.28·10-s + 1.46i·11-s − 0.288·12-s − 0.184·14-s + 1.04i·15-s + 0.250·16-s − 1.79·17-s − 0.235i·18-s + ⋯ |
Λ(s)=(=(1014s/2ΓC(s)L(s)(0.0304−0.999i)Λ(2−s)
Λ(s)=(=(1014s/2ΓC(s+1/2)L(s)(0.0304−0.999i)Λ(1−s)
Degree: |
2 |
Conductor: |
1014
= 2⋅3⋅132
|
Sign: |
0.0304−0.999i
|
Analytic conductor: |
8.09683 |
Root analytic conductor: |
2.84549 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1014(337,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1014, ( :1/2), 0.0304−0.999i)
|
Particular Values
L(1) |
≈ |
1.311680798 |
L(21) |
≈ |
1.311680798 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+iT |
| 3 | 1−T |
| 13 | 1 |
good | 5 | 1−4.04iT−5T2 |
| 7 | 1+0.692iT−7T2 |
| 11 | 1−4.85iT−11T2 |
| 17 | 1+7.38T+17T2 |
| 19 | 1+1.78iT−19T2 |
| 23 | 1+5.10T+23T2 |
| 29 | 1+3.34T+29T2 |
| 31 | 1−0.972iT−31T2 |
| 37 | 1+1.28iT−37T2 |
| 41 | 1−1.50iT−41T2 |
| 43 | 1−8.31T+43T2 |
| 47 | 1−7.20iT−47T2 |
| 53 | 1−13.4T+53T2 |
| 59 | 1+1.30iT−59T2 |
| 61 | 1+0.396T+61T2 |
| 67 | 1−6.05iT−67T2 |
| 71 | 1+1.32iT−71T2 |
| 73 | 1−7.65iT−73T2 |
| 79 | 1+8.33T+79T2 |
| 83 | 1−15.3iT−83T2 |
| 89 | 1+3.10iT−89T2 |
| 97 | 1+8.54iT−97T2 |
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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
−10.18309634630888290225387004116, −9.624030967868472264902246746072, −8.666359133629152854343203103407, −7.38757951516034581991644915916, −7.10649551024888338897072314165, −6.05075647708314483753185269432, −4.44576254840836318438154960731, −3.82253387853839169425188470607, −2.54203323910779082045144166718, −2.13397469367192747351334117366,
0.53068843562592457290458618166, 2.06510733659147620384261976194, 3.77955172827098274832144802424, 4.48838879773782674869707215619, 5.53731470480394207642060446762, 6.13491554949825851366793948114, 7.47198525089832125550909266730, 8.365398230448837643681388581185, 8.798267522111578953997592293996, 9.197688213805736506823808132532