L(s) = 1 | + 2·2-s + 3·4-s − 7-s + 4·8-s − 2·9-s − 2·14-s + 5·16-s − 4·18-s − 3·28-s − 12·29-s + 6·32-s − 6·36-s − 4·37-s − 16·43-s + 49-s − 12·53-s − 4·56-s − 24·58-s + 2·63-s + 7·64-s + 8·67-s − 8·72-s − 8·74-s + 16·79-s − 5·81-s − 32·86-s + 2·98-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.377·7-s + 1.41·8-s − 2/3·9-s − 0.534·14-s + 5/4·16-s − 0.942·18-s − 0.566·28-s − 2.22·29-s + 1.06·32-s − 36-s − 0.657·37-s − 2.43·43-s + 1/7·49-s − 1.64·53-s − 0.534·56-s − 3.15·58-s + 0.251·63-s + 7/8·64-s + 0.977·67-s − 0.942·72-s − 0.929·74-s + 1.80·79-s − 5/9·81-s − 3.45·86-s + 0.202·98-s + ⋯ |
Λ(s)=(=(857500s/2ΓC(s)2L(s)−Λ(2−s)
Λ(s)=(=(857500s/2ΓC(s+1/2)2L(s)−Λ(1−s)
Degree: |
4 |
Conductor: |
857500
= 22⋅54⋅73
|
Sign: |
−1
|
Analytic conductor: |
54.6749 |
Root analytic conductor: |
2.71923 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
1
|
Selberg data: |
(4, 857500, ( :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 |
| 7 | C1 | 1+T |
good | 3 | C2 | (1−2T+pT2)(1+2T+pT2) |
| 11 | C2 | (1+pT2)2 |
| 13 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 17 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 19 | C2 | (1−2T+pT2)(1+2T+pT2) |
| 23 | C2 | (1+pT2)2 |
| 29 | C2 | (1+6T+pT2)2 |
| 31 | C2 | (1−4T+pT2)(1+4T+pT2) |
| 37 | C2 | (1+2T+pT2)2 |
| 41 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 43 | C2 | (1+8T+pT2)2 |
| 47 | C2 | (1−12T+pT2)(1+12T+pT2) |
| 53 | C2 | (1+6T+pT2)2 |
| 59 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 61 | C2 | (1−8T+pT2)(1+8T+pT2) |
| 67 | C2 | (1−4T+pT2)2 |
| 71 | C2 | (1+pT2)2 |
| 73 | C2 | (1−2T+pT2)(1+2T+pT2) |
| 79 | C2 | (1−8T+pT2)2 |
| 83 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 89 | C2 | (1−6T+pT2)(1+6T+pT2) |
| 97 | C2 | (1−10T+pT2)(1+10T+pT2) |
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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
−7.71895664384636679391588711627, −7.60556812848957435561623382149, −6.73571587686584923478967639363, −6.66965130292041403123447657278, −6.15190325448345686265108903037, −5.59857578974307577358363282055, −5.12614454716073682582164065811, −5.04607295657079957887442801881, −4.11500666640753372808858013263, −3.77049860941673777607586608067, −3.28327332646103144207952017478, −2.83590121110111537782644859852, −2.07204529193989626181377807359, −1.54150783577442522625509630661, 0,
1.54150783577442522625509630661, 2.07204529193989626181377807359, 2.83590121110111537782644859852, 3.28327332646103144207952017478, 3.77049860941673777607586608067, 4.11500666640753372808858013263, 5.04607295657079957887442801881, 5.12614454716073682582164065811, 5.59857578974307577358363282055, 6.15190325448345686265108903037, 6.66965130292041403123447657278, 6.73571587686584923478967639363, 7.60556812848957435561623382149, 7.71895664384636679391588711627