L(s) = 1 | − 8·2-s + 48·4-s − 53·5-s − 256·8-s + 424·10-s − 191·11-s + 379·13-s + 1.28e3·16-s − 340·17-s + 1.76e3·19-s − 2.54e3·20-s + 1.52e3·22-s − 3.23e3·23-s − 1.74e3·25-s − 3.03e3·26-s − 4.45e3·29-s − 1.99e3·31-s − 6.14e3·32-s + 2.72e3·34-s + 2.05e4·37-s − 1.41e4·38-s + 1.35e4·40-s + 8.81e3·41-s + 1.58e4·43-s − 9.16e3·44-s + 2.58e4·46-s + 3.39e4·47-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.948·5-s − 1.41·8-s + 1.34·10-s − 0.475·11-s + 0.621·13-s + 5/4·16-s − 0.285·17-s + 1.12·19-s − 1.42·20-s + 0.673·22-s − 1.27·23-s − 0.557·25-s − 0.879·26-s − 0.984·29-s − 0.372·31-s − 1.06·32-s + 0.403·34-s + 2.47·37-s − 1.58·38-s + 1.34·40-s + 0.818·41-s + 1.30·43-s − 0.713·44-s + 1.80·46-s + 2.23·47-s + ⋯ |
Λ(s)=(=(777924s/2ΓC(s)2L(s)Λ(6−s)
Λ(s)=(=(777924s/2ΓC(s+5/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
777924
= 22⋅34⋅74
|
Sign: |
1
|
Analytic conductor: |
20010.5 |
Root analytic conductor: |
11.8936 |
Motivic weight: |
5 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
2
|
Selberg data: |
(4, 777924, ( :5/2,5/2), 1)
|
Particular Values
L(3) |
= |
0 |
L(21) |
= |
0 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | C1 | (1+p2T)2 |
| 3 | | 1 |
| 7 | | 1 |
good | 5 | D4 | 1+53T+4552T2+53p5T3+p10T4 |
| 11 | D4 | 1+191T+24656pT2+191p5T3+p10T4 |
| 13 | D4 | 1−379T+488066T2−379p5T3+p10T4 |
| 17 | D4 | 1+20pT+1908514T2+20p6T3+p10T4 |
| 19 | D4 | 1−1769T+2083758T2−1769p5T3+p10T4 |
| 23 | D4 | 1+3236T+15452206T2+3236p5T3+p10T4 |
| 29 | D4 | 1+4459T+37061638T2+4459p5T3+p10T4 |
| 31 | D4 | 1+1994T−6304813T2+1994p5T3+p10T4 |
| 37 | D4 | 1−20587T+238401006T2−20587p5T3+p10T4 |
| 41 | D4 | 1−8814T+240678562T2−8814p5T3+p10T4 |
| 43 | D4 | 1−15853T+338678796T2−15853p5T3+p10T4 |
| 47 | D4 | 1−33912T+687783466T2−33912p5T3+p10T4 |
| 53 | D4 | 1+49239T+1320998110T2+49239p5T3+p10T4 |
| 59 | D4 | 1+56735T+2230528834T2+56735p5T3+p10T4 |
| 61 | D4 | 1+67508T+2826067262T2+67508p5T3+p10T4 |
| 67 | D4 | 1−75723T+3861149404T2−75723p5T3+p10T4 |
| 71 | D4 | 1−8992T−681216182T2−8992p5T3+p10T4 |
| 73 | D4 | 1−3201T+2311013380T2−3201p5T3+p10T4 |
| 79 | D4 | 1−26612T+3997648985T2−26612p5T3+p10T4 |
| 83 | D4 | 1+949T+367057696T2+949p5T3+p10T4 |
| 89 | D4 | 1−176562T+18855802834T2−176562p5T3+p10T4 |
| 97 | D4 | 1−129423T+11942811256T2−129423p5T3+p10T4 |
show more | | |
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L(s)=p∏ j=1∏4(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−9.122967579408345301105672359664, −9.028836540491247939619092079475, −8.055112854525507532072218935418, −7.87488213768432862565109879127, −7.60495344669563393371368526228, −7.59986802901357946926291486359, −6.70346829860673560579288455765, −6.18638740186039916205990629628, −5.88171352231280897524639204479, −5.51679574437906724904096269878, −4.49425724696113673905834650185, −4.33068137825321978997266489929, −3.50311526236590679312837458846, −3.30090166998639905152420973040, −2.33876026178801805805615635408, −2.21931967658440275607589337776, −1.21058453152264209795029917407, −0.946722620384081625664183226747, 0, 0,
0.946722620384081625664183226747, 1.21058453152264209795029917407, 2.21931967658440275607589337776, 2.33876026178801805805615635408, 3.30090166998639905152420973040, 3.50311526236590679312837458846, 4.33068137825321978997266489929, 4.49425724696113673905834650185, 5.51679574437906724904096269878, 5.88171352231280897524639204479, 6.18638740186039916205990629628, 6.70346829860673560579288455765, 7.59986802901357946926291486359, 7.60495344669563393371368526228, 7.87488213768432862565109879127, 8.055112854525507532072218935418, 9.028836540491247939619092079475, 9.122967579408345301105672359664