L(s) = 1 | + 20·2-s + 1.64e3·4-s + 6.25e3·5-s + 5.79e4·7-s + 9.85e4·8-s + 1.25e5·10-s + 6.18e5·11-s + 3.41e6·13-s + 1.15e6·14-s + 6.29e5·16-s − 1.31e6·17-s + 5.32e6·19-s + 1.02e7·20-s + 1.23e7·22-s − 5.89e7·23-s + 2.92e7·25-s + 6.82e7·26-s + 9.49e7·28-s − 9.41e7·29-s + 2.44e8·31-s + 1.18e8·32-s − 2.63e7·34-s + 3.61e8·35-s + 2.10e7·37-s + 1.06e8·38-s + 6.16e8·40-s + 7.45e8·41-s + ⋯ |
L(s) = 1 | + 0.441·2-s + 0.800·4-s + 0.894·5-s + 1.30·7-s + 1.06·8-s + 0.395·10-s + 1.15·11-s + 2.55·13-s + 0.575·14-s + 0.150·16-s − 0.225·17-s + 0.493·19-s + 0.716·20-s + 0.511·22-s − 1.90·23-s + 3/5·25-s + 1.12·26-s + 1.04·28-s − 0.852·29-s + 1.53·31-s + 0.622·32-s − 0.0994·34-s + 1.16·35-s + 0.0497·37-s + 0.218·38-s + 0.951·40-s + 1.00·41-s + ⋯ |
Λ(s)=(=(2025s/2ΓC(s)2L(s)Λ(12−s)
Λ(s)=(=(2025s/2ΓC(s+11/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
2025
= 34⋅52
|
Sign: |
1
|
Analytic conductor: |
1195.46 |
Root analytic conductor: |
5.88008 |
Motivic weight: |
11 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 2025, ( :11/2,11/2), 1)
|
Particular Values
L(6) |
≈ |
10.39862471 |
L(21) |
≈ |
10.39862471 |
L(213) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 3 | | 1 |
| 5 | C1 | (1−p5T)2 |
good | 2 | D4 | 1−5p2T−155p3T2−5p13T3+p22T4 |
| 7 | D4 | 1−57900T+660624350pT2−57900p11T3+p22T4 |
| 11 | D4 | 1−618176T+245194892966T2−618176p11T3+p22T4 |
| 13 | D4 | 1−3414260T+6398662197390T2−3414260p11T3+p22T4 |
| 17 | D4 | 1+1317940T+59308395866630T2+1317940p11T3+p22T4 |
| 19 | D4 | 1−280280pT+194538827137638T2−280280p12T3+p22T4 |
| 23 | D4 | 1+2562780pT+2773540471931410T2+2562780p12T3+p22T4 |
| 29 | D4 | 1+3246220pT+23426350431097358T2+3246220p12T3+p22T4 |
| 31 | D4 | 1−244543464T+34393316207729486T2−244543464p11T3+p22T4 |
| 37 | D4 | 1−21003220T+137126715218410590T2−21003220p11T3+p22T4 |
| 41 | D4 | 1−745743316T+929792912462405846T2−745743316p11T3+p22T4 |
| 43 | D4 | 1−629950100T+1840945003918927050T2−629950100p11T3+p22T4 |
| 47 | D4 | 1−1402061540T+5181805952108806370T2−1402061540p11T3+p22T4 |
| 53 | D4 | 1+1138320580T−2203723231625575330T2+1138320580p11T3+p22T4 |
| 59 | D4 | 1+7317515560T+55027608950440780118T2+7317515560p11T3+p22T4 |
| 61 | D4 | 1+1516425676T+85869525433683691566T2+1516425676p11T3+p22T4 |
| 67 | D4 | 1−15734290140T+30⋯30T2−15734290140p11T3+p22T4 |
| 71 | D4 | 1+32938471544T+73⋯26T2+32938471544p11T3+p22T4 |
| 73 | D4 | 1+29982848860T+78⋯10T2+29982848860p11T3+p22T4 |
| 79 | D4 | 1+3302823120T+12⋯58T2+3302823120p11T3+p22T4 |
| 83 | D4 | 1+13299102420T+18⋯30T2+13299102420p11T3+p22T4 |
| 89 | D4 | 1−12674770860T+43⋯78T2−12674770860p11T3+p22T4 |
| 97 | D4 | 1+3080703740T+18⋯70T2+3080703740p11T3+p22T4 |
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
−13.59106970106290456839370614740, −13.56710128098156843287499157000, −12.55223479424372090912921781289, −11.69758221571725753929748401754, −11.40418902321999187145402067319, −10.90304658380817722644706725155, −10.36095623571164980771795751486, −9.483558352005119969532702819207, −8.773033566273907711214682152651, −8.150800460180734116353941224084, −7.52842368174729526223788513249, −6.50372507392322184884428906988, −6.12918124445120266732680931286, −5.58109298499102332828255387665, −4.32419327711743112445260313694, −4.16813968501628161083361706858, −3.00267797686282288936505338766, −1.83037914699779268756640265577, −1.60277424795465373480873090605, −0.986321171235488763153926762507,
0.986321171235488763153926762507, 1.60277424795465373480873090605, 1.83037914699779268756640265577, 3.00267797686282288936505338766, 4.16813968501628161083361706858, 4.32419327711743112445260313694, 5.58109298499102332828255387665, 6.12918124445120266732680931286, 6.50372507392322184884428906988, 7.52842368174729526223788513249, 8.150800460180734116353941224084, 8.773033566273907711214682152651, 9.483558352005119969532702819207, 10.36095623571164980771795751486, 10.90304658380817722644706725155, 11.40418902321999187145402067319, 11.69758221571725753929748401754, 12.55223479424372090912921781289, 13.56710128098156843287499157000, 13.59106970106290456839370614740