L(s) = 1 | + 2·2-s − 3-s + 3·4-s − 2·6-s − 5·7-s + 4·8-s − 2·9-s − 2·11-s − 3·12-s − 9·13-s − 10·14-s + 5·16-s − 3·17-s − 4·18-s + 6·19-s + 5·21-s − 4·22-s − 7·23-s − 4·24-s − 18·26-s + 2·27-s − 15·28-s + 2·29-s − 5·31-s + 6·32-s + 2·33-s − 6·34-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 0.577·3-s + 3/2·4-s − 0.816·6-s − 1.88·7-s + 1.41·8-s − 2/3·9-s − 0.603·11-s − 0.866·12-s − 2.49·13-s − 2.67·14-s + 5/4·16-s − 0.727·17-s − 0.942·18-s + 1.37·19-s + 1.09·21-s − 0.852·22-s − 1.45·23-s − 0.816·24-s − 3.53·26-s + 0.384·27-s − 2.83·28-s + 0.371·29-s − 0.898·31-s + 1.06·32-s + 0.348·33-s − 1.02·34-s + ⋯ |
Λ(s)=(=(2102500s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(2102500s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
2102500
= 22⋅54⋅292
|
Sign: |
1
|
Analytic conductor: |
134.057 |
Root analytic conductor: |
3.40269 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
2
|
Selberg data: |
(4, 2102500, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
= |
0 |
L(21) |
= |
0 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | C1 | (1−T)2 |
| 5 | | 1 |
| 29 | C1 | (1−T)2 |
good | 3 | D4 | 1+T+pT2+pT3+p2T4 |
| 7 | D4 | 1+5T+17T2+5pT3+p2T4 |
| 11 | D4 | 1+2T+10T2+2pT3+p2T4 |
| 13 | D4 | 1+9T+43T2+9pT3+p2T4 |
| 17 | D4 | 1+3T+7T2+3pT3+p2T4 |
| 19 | D4 | 1−6T+34T2−6pT3+p2T4 |
| 23 | D4 | 1+7T+55T2+7pT3+p2T4 |
| 31 | D4 | 1+5T+39T2+5pT3+p2T4 |
| 37 | D4 | 1−2T−42T2−2pT3+p2T4 |
| 41 | D4 | 1+10T+94T2+10pT3+p2T4 |
| 43 | D4 | 1+3T+85T2+3pT3+p2T4 |
| 47 | C2 | (1+pT2)2 |
| 53 | D4 | 1+3T+79T2+3pT3+p2T4 |
| 59 | D4 | 1−9T+109T2−9pT3+p2T4 |
| 61 | D4 | 1+17T+165T2+17pT3+p2T4 |
| 67 | D4 | 1−4T+86T2−4pT3+p2T4 |
| 71 | C2 | (1+pT2)2 |
| 73 | D4 | 1−13T+107T2−13pT3+p2T4 |
| 79 | D4 | 1−T+129T2−pT3+p2T4 |
| 83 | D4 | 1−10T+178T2−10pT3+p2T4 |
| 89 | D4 | 1+22T+286T2+22pT3+p2T4 |
| 97 | D4 | 1+13T+207T2+13pT3+p2T4 |
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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
−9.407990081614753518438997617253, −9.349563612736162688935347605120, −8.157589861722829361886774702904, −8.122977075012076541251019641000, −7.52883759681880672876433442106, −6.99003145307253373700181367280, −6.70473469805233267182206547619, −6.56078543410263305001936796313, −5.84236168494095225366956922217, −5.49718281366288437666415243477, −5.22680105345781307133436841878, −4.87870545989729162912470542411, −4.18060447012509745400533690702, −3.78509387676597826168849823887, −3.12032659178808281312473073020, −2.77834456180461400016046101889, −2.50474533305269970826657323526, −1.69465286739611247746851171454, 0, 0,
1.69465286739611247746851171454, 2.50474533305269970826657323526, 2.77834456180461400016046101889, 3.12032659178808281312473073020, 3.78509387676597826168849823887, 4.18060447012509745400533690702, 4.87870545989729162912470542411, 5.22680105345781307133436841878, 5.49718281366288437666415243477, 5.84236168494095225366956922217, 6.56078543410263305001936796313, 6.70473469805233267182206547619, 6.99003145307253373700181367280, 7.52883759681880672876433442106, 8.122977075012076541251019641000, 8.157589861722829361886774702904, 9.349563612736162688935347605120, 9.407990081614753518438997617253