L(s) = 1 | + 80·2-s − 3.60e3·4-s + 3.12e4·5-s − 6.16e5·7-s − 4.32e5·8-s + 2.50e6·10-s + 2.46e6·11-s + 6.51e6·13-s − 4.93e7·14-s − 1.32e7·16-s − 6.33e5·17-s − 3.74e8·19-s − 1.12e8·20-s + 1.97e8·22-s − 6.21e8·23-s + 7.32e8·25-s + 5.21e8·26-s + 2.21e9·28-s + 7.79e9·29-s − 4.95e9·31-s − 4.46e9·32-s − 5.06e7·34-s − 1.92e10·35-s + 2.18e10·37-s − 2.99e10·38-s − 1.35e10·40-s − 5.56e9·41-s + ⋯ |
L(s) = 1 | + 0.883·2-s − 0.439·4-s + 0.894·5-s − 1.97·7-s − 0.583·8-s + 0.790·10-s + 0.419·11-s + 0.374·13-s − 1.75·14-s − 0.198·16-s − 0.00636·17-s − 1.82·19-s − 0.393·20-s + 0.371·22-s − 0.876·23-s + 3/5·25-s + 0.331·26-s + 0.870·28-s + 2.43·29-s − 1.00·31-s − 0.735·32-s − 0.00562·34-s − 1.77·35-s + 1.39·37-s − 1.61·38-s − 0.521·40-s − 0.182·41-s + ⋯ |
Λ(s)=(=(2025s/2ΓC(s)2L(s)Λ(14−s)
Λ(s)=(=(2025s/2ΓC(s+13/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
2025
= 34⋅52
|
Sign: |
1
|
Analytic conductor: |
2328.44 |
Root analytic conductor: |
6.94650 |
Motivic weight: |
13 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
2
|
Selberg data: |
(4, 2025, ( :13/2,13/2), 1)
|
Particular Values
L(7) |
= |
0 |
L(21) |
= |
0 |
L(215) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 3 | | 1 |
| 5 | C1 | (1−p6T)2 |
good | 2 | D4 | 1−5p4T+625p4T2−5p17T3+p26T4 |
| 7 | D4 | 1+616300T+38195042750pT2+616300p13T3+p26T4 |
| 11 | D4 | 1−2467136T+1885360936226pT2−2467136p13T3+p26T4 |
| 13 | D4 | 1−6517860T+474444414946750T2−6517860p13T3+p26T4 |
| 17 | D4 | 1+633460T+11487320401034950T2+633460p13T3+p26T4 |
| 19 | D4 | 1+374063400T+6186197213067922pT2+374063400p13T3+p26T4 |
| 23 | D4 | 1+621982140T+381235795435716850T2+621982140p13T3+p26T4 |
| 29 | D4 | 1−7795134100T+33093039872067713278T2−7795134100p13T3+p26T4 |
| 31 | D4 | 1+4957819816T+26301842446694653646T2+4957819816p13T3+p26T4 |
| 37 | D4 | 1−21833071780T+28⋯50T2−21833071780p13T3+p26T4 |
| 41 | D4 | 1+5565813644T+18⋯26T2+5565813644p13T3+p26T4 |
| 43 | D4 | 1+52510877700T+41⋯50T2+52510877700p13T3+p26T4 |
| 47 | D4 | 1+92855886340T+12⋯50T2+92855886340p13T3+p26T4 |
| 53 | D4 | 1+266248876180T+64⋯50T2+266248876180p13T3+p26T4 |
| 59 | D4 | 1+253501607800T+16⋯58T2+253501607800p13T3+p26T4 |
| 61 | D4 | 1+377459239836T+32⋯86T2+377459239836p13T3+p26T4 |
| 67 | D4 | 1+2375782313740T+25⋯50T2+2375782313740p13T3+p26T4 |
| 71 | D4 | 1−556190102776T+15⋯66T2−556190102776p13T3+p26T4 |
| 73 | D4 | 1+556465382460T+29⋯50T2+556465382460p13T3+p26T4 |
| 79 | D4 | 1+2408230567600T+44⋯78T2+2408230567600p13T3+p26T4 |
| 83 | D4 | 1−3898295602980T+20⋯50T2−3898295602980p13T3+p26T4 |
| 89 | D4 | 1+106847675700T+43⋯38T2+106847675700p13T3+p26T4 |
| 97 | D4 | 1−18744409451140T+21⋯50T2−18744409451140p13T3+p26T4 |
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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
−12.73591540311249138851913966273, −12.63399231470785806222527772929, −11.78104840321772313237440450521, −10.83518279080862306124119364239, −10.07114919878083405459717323229, −9.937062189600086689558788208416, −8.954454909570612616119694337647, −8.854294366333990492379938197363, −7.74897356819101288692771710774, −6.57201332815399923541023244790, −6.22814505082278178330457576962, −6.08886496033399923850353020686, −4.74357319851420580747765435389, −4.51049237935660213249048214948, −3.48266709739154217327358336712, −3.11950391507079835013279077268, −2.17265478868093070678338155590, −1.32266173339432277477382555465, 0, 0,
1.32266173339432277477382555465, 2.17265478868093070678338155590, 3.11950391507079835013279077268, 3.48266709739154217327358336712, 4.51049237935660213249048214948, 4.74357319851420580747765435389, 6.08886496033399923850353020686, 6.22814505082278178330457576962, 6.57201332815399923541023244790, 7.74897356819101288692771710774, 8.854294366333990492379938197363, 8.954454909570612616119694337647, 9.937062189600086689558788208416, 10.07114919878083405459717323229, 10.83518279080862306124119364239, 11.78104840321772313237440450521, 12.63399231470785806222527772929, 12.73591540311249138851913966273