L(s) = 1 | + (−3.31 − 1.37i)5-s + (−3.66 + 1.51i)7-s + (0.147 + 0.355i)11-s + 1.23i·13-s + (4.06 + 0.666i)17-s + (−0.0317 + 0.0317i)19-s + (−2.04 − 4.93i)23-s + (5.57 + 5.57i)25-s + (4.62 + 1.91i)29-s + (0.946 − 2.28i)31-s + 14.2·35-s + (1.90 − 4.59i)37-s + (−1.75 + 0.728i)41-s + (6.76 + 6.76i)43-s − 2.73i·47-s + ⋯ |
L(s) = 1 | + (−1.48 − 0.614i)5-s + (−1.38 + 0.574i)7-s + (0.0444 + 0.107i)11-s + 0.342i·13-s + (0.986 + 0.161i)17-s + (−0.00727 + 0.00727i)19-s + (−0.426 − 1.02i)23-s + (1.11 + 1.11i)25-s + (0.858 + 0.355i)29-s + (0.170 − 0.410i)31-s + 2.40·35-s + (0.312 − 0.755i)37-s + (−0.274 + 0.113i)41-s + (1.03 + 1.03i)43-s − 0.398i·47-s + ⋯ |
Λ(s)=(=(1224s/2ΓC(s)L(s)(0.996+0.0863i)Λ(2−s)
Λ(s)=(=(1224s/2ΓC(s+1/2)L(s)(0.996+0.0863i)Λ(1−s)
Degree: |
2 |
Conductor: |
1224
= 23⋅32⋅17
|
Sign: |
0.996+0.0863i
|
Analytic conductor: |
9.77368 |
Root analytic conductor: |
3.12629 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1224(433,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1224, ( :1/2), 0.996+0.0863i)
|
Particular Values
L(1) |
≈ |
0.8669774929 |
L(21) |
≈ |
0.8669774929 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 17 | 1+(−4.06−0.666i)T |
good | 5 | 1+(3.31+1.37i)T+(3.53+3.53i)T2 |
| 7 | 1+(3.66−1.51i)T+(4.94−4.94i)T2 |
| 11 | 1+(−0.147−0.355i)T+(−7.77+7.77i)T2 |
| 13 | 1−1.23iT−13T2 |
| 19 | 1+(0.0317−0.0317i)T−19iT2 |
| 23 | 1+(2.04+4.93i)T+(−16.2+16.2i)T2 |
| 29 | 1+(−4.62−1.91i)T+(20.5+20.5i)T2 |
| 31 | 1+(−0.946+2.28i)T+(−21.9−21.9i)T2 |
| 37 | 1+(−1.90+4.59i)T+(−26.1−26.1i)T2 |
| 41 | 1+(1.75−0.728i)T+(28.9−28.9i)T2 |
| 43 | 1+(−6.76−6.76i)T+43iT2 |
| 47 | 1+2.73iT−47T2 |
| 53 | 1+(−6.87+6.87i)T−53iT2 |
| 59 | 1+(−7.58−7.58i)T+59iT2 |
| 61 | 1+(−12.3+5.12i)T+(43.1−43.1i)T2 |
| 67 | 1+4.69T+67T2 |
| 71 | 1+(3.55−8.57i)T+(−50.2−50.2i)T2 |
| 73 | 1+(5.02+2.08i)T+(51.6+51.6i)T2 |
| 79 | 1+(1.58+3.83i)T+(−55.8+55.8i)T2 |
| 83 | 1+(1.37−1.37i)T−83iT2 |
| 89 | 1−10.9iT−89T2 |
| 97 | 1+(−6.88−2.85i)T+(68.5+68.5i)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
−9.661336017730781098934200475294, −8.783758344797768336761039762162, −8.200549112758727790541170307130, −7.28158748653232634076909275820, −6.45543473717053227349773819118, −5.49849753783661670796755010502, −4.34906003870503064340188799263, −3.65195266884981260970640982326, −2.64296283313921754951680727790, −0.67490892199528815424594477083,
0.67328054559012428692933793527, 2.91186936746385387183974146520, 3.51747951205868498264853709023, 4.23995536016906616789907982626, 5.63184005827182356534136074786, 6.61943587179762728848537365173, 7.32019312466692670224369987061, 7.88388947285219507636795221769, 8.870551574365622599499674640522, 10.02661203129232837548338731603