L(s) = 1 | − 2·2-s + 5·5-s + 4·7-s + 8·8-s − 10·10-s + 48·11-s − 2·13-s − 8·14-s − 16·16-s − 228·17-s + 280·19-s − 96·22-s − 72·23-s + 4·26-s − 210·29-s − 272·31-s + 456·34-s + 20·35-s − 668·37-s − 560·38-s + 40·40-s + 198·41-s + 268·43-s + 144·46-s − 216·47-s + 343·49-s − 156·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.447·5-s + 0.215·7-s + 0.353·8-s − 0.316·10-s + 1.31·11-s − 0.0426·13-s − 0.152·14-s − 1/4·16-s − 3.25·17-s + 3.38·19-s − 0.930·22-s − 0.652·23-s + 0.0301·26-s − 1.34·29-s − 1.57·31-s + 2.30·34-s + 0.0965·35-s − 2.96·37-s − 2.39·38-s + 0.158·40-s + 0.754·41-s + 0.950·43-s + 0.461·46-s − 0.670·47-s + 49-s − 0.404·53-s + ⋯ |
Λ(s)=(=(656100s/2ΓC(s)2L(s)Λ(4−s)
Λ(s)=(=(656100s/2ΓC(s+3/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
656100
= 22⋅38⋅52
|
Sign: |
1
|
Analytic conductor: |
2284.03 |
Root analytic conductor: |
6.91314 |
Motivic weight: |
3 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 656100, ( :3/2,3/2), 1)
|
Particular Values
L(2) |
≈ |
0.4868536288 |
L(21) |
≈ |
0.4868536288 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | C2 | 1+pT+p2T2 |
| 3 | | 1 |
| 5 | C2 | 1−pT+p2T2 |
good | 7 | C22 | 1−4T−327T2−4p3T3+p6T4 |
| 11 | C22 | 1−48T+973T2−48p3T3+p6T4 |
| 13 | C22 | 1+2T−2193T2+2p3T3+p6T4 |
| 17 | C2 | (1+114T+p3T2)2 |
| 19 | C2 | (1−140T+p3T2)2 |
| 23 | C22 | 1+72T−6983T2+72p3T3+p6T4 |
| 29 | C22 | 1+210T+19711T2+210p3T3+p6T4 |
| 31 | C22 | 1+272T+44193T2+272p3T3+p6T4 |
| 37 | C2 | (1+334T+p3T2)2 |
| 41 | C22 | 1−198T−29717T2−198p3T3+p6T4 |
| 43 | C22 | 1−268T−7683T2−268p3T3+p6T4 |
| 47 | C22 | 1+216T−57167T2+216p3T3+p6T4 |
| 53 | C2 | (1+78T+p3T2)2 |
| 59 | C22 | 1+240T−147779T2+240p3T3+p6T4 |
| 61 | C22 | 1+302T−135777T2+302p3T3+p6T4 |
| 67 | C22 | 1+596T+54453T2+596p3T3+p6T4 |
| 71 | C2 | (1+768T+p3T2)2 |
| 73 | C2 | (1+478T+p3T2)2 |
| 79 | C22 | 1−640T−83439T2−640p3T3+p6T4 |
| 83 | C22 | 1−348T−450683T2−348p3T3+p6T4 |
| 89 | C2 | (1−210T+p3T2)2 |
| 97 | C22 | 1−1534T+1440483T2−1534p3T3+p6T4 |
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
−9.948803142664595719246401922463, −9.294011631793388166509074229030, −9.250988954048808336708468966002, −8.984159280521740295314032792267, −8.737245772324103545045651916639, −7.81110752509252653158896263146, −7.53731043249234003750357538343, −7.04557397410445080895579418777, −6.85223204447930698287103087803, −6.20675495835752627143755411642, −5.52004068151849288517290720147, −5.42876452954204143358306610172, −4.59802095882482836037123259004, −4.21461360619803631157018387916, −3.63328139636967747284214443138, −3.09250217730309355128884354598, −2.23787650472173326400753020356, −1.59847307293441694674770582672, −1.39266072303511398267761898096, −0.20970422872946751674271682152,
0.20970422872946751674271682152, 1.39266072303511398267761898096, 1.59847307293441694674770582672, 2.23787650472173326400753020356, 3.09250217730309355128884354598, 3.63328139636967747284214443138, 4.21461360619803631157018387916, 4.59802095882482836037123259004, 5.42876452954204143358306610172, 5.52004068151849288517290720147, 6.20675495835752627143755411642, 6.85223204447930698287103087803, 7.04557397410445080895579418777, 7.53731043249234003750357538343, 7.81110752509252653158896263146, 8.737245772324103545045651916639, 8.984159280521740295314032792267, 9.250988954048808336708468966002, 9.294011631793388166509074229030, 9.948803142664595719246401922463