L(s) = 1 | + 2·2-s + 2·4-s − 2·5-s + 5·9-s − 4·10-s − 4·11-s + 6·13-s − 4·16-s + 6·17-s + 10·18-s − 4·20-s − 8·22-s − 12·23-s − 7·25-s + 12·26-s − 8·32-s + 12·34-s + 10·36-s − 6·37-s − 8·44-s − 10·45-s − 24·46-s + 5·49-s − 14·50-s + 12·52-s + 8·55-s − 20·59-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 4-s − 0.894·5-s + 5/3·9-s − 1.26·10-s − 1.20·11-s + 1.66·13-s − 16-s + 1.45·17-s + 2.35·18-s − 0.894·20-s − 1.70·22-s − 2.50·23-s − 7/5·25-s + 2.35·26-s − 1.41·32-s + 2.05·34-s + 5/3·36-s − 0.986·37-s − 1.20·44-s − 1.49·45-s − 3.53·46-s + 5/7·49-s − 1.97·50-s + 1.66·52-s + 1.07·55-s − 2.60·59-s + ⋯ |
Λ(s)=(=(10816s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(10816s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
10816
= 26⋅132
|
Sign: |
1
|
Analytic conductor: |
0.689637 |
Root analytic conductor: |
0.911287 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 10816, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
1.804631911 |
L(21) |
≈ |
1.804631911 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | C2 | 1−pT+pT2 |
| 13 | C2 | 1−6T+pT2 |
good | 3 | C22 | 1−5T2+p2T4 |
| 5 | C2 | (1+T+pT2)2 |
| 7 | C22 | 1−5T2+p2T4 |
| 11 | C2 | (1+2T+pT2)2 |
| 17 | C2 | (1−3T+pT2)2 |
| 19 | C2 | (1+pT2)2 |
| 23 | C2 | (1+6T+pT2)2 |
| 29 | C22 | 1−22T2+p2T4 |
| 31 | C2 | (1−pT2)2 |
| 37 | C2 | (1+3T+pT2)2 |
| 41 | C2 | (1−8T+pT2)(1+8T+pT2) |
| 43 | C22 | 1−5T2+p2T4 |
| 47 | C22 | 1−45T2+p2T4 |
| 53 | C22 | 1−70T2+p2T4 |
| 59 | C2 | (1+10T+pT2)2 |
| 61 | C2 | (1−12T+pT2)(1+12T+pT2) |
| 67 | C2 | (1−12T+pT2)2 |
| 71 | C22 | 1−117T2+p2T4 |
| 73 | C2 | (1−16T+pT2)(1+16T+pT2) |
| 79 | C2 | (1+pT2)2 |
| 83 | C2 | (1−16T+pT2)2 |
| 89 | C22 | 1−162T2+p2T4 |
| 97 | C2 | (1−8T+pT2)(1+8T+pT2) |
show more | | |
show less | | |
L(s)=p∏ j=1∏4(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−13.95530504425463177219684818725, −13.37173768154440220279298237117, −13.27491948009287716451620243992, −12.36455323142233149799899267619, −12.13823934744062902284794236554, −11.87880212660656222739612614403, −10.84737676313181404484593279239, −10.63166485054981770210982109794, −9.858004989560526093512736265896, −9.416061035242936798259608912489, −8.110565670284822805233160735207, −8.042626235264431202428795382995, −7.41462406189326598592223265204, −6.51233162850101765397403798126, −5.92452216695234809845577081561, −5.33318281045966834656803326883, −4.43460414278886521159715690280, −3.72887635245383174516693223809, −3.59233905038738281641359546513, −1.97087923560232949779890484285,
1.97087923560232949779890484285, 3.59233905038738281641359546513, 3.72887635245383174516693223809, 4.43460414278886521159715690280, 5.33318281045966834656803326883, 5.92452216695234809845577081561, 6.51233162850101765397403798126, 7.41462406189326598592223265204, 8.042626235264431202428795382995, 8.110565670284822805233160735207, 9.416061035242936798259608912489, 9.858004989560526093512736265896, 10.63166485054981770210982109794, 10.84737676313181404484593279239, 11.87880212660656222739612614403, 12.13823934744062902284794236554, 12.36455323142233149799899267619, 13.27491948009287716451620243992, 13.37173768154440220279298237117, 13.95530504425463177219684818725