L(s) = 1 | + 2·2-s − 14·3-s − 4-s − 28·6-s + 28·7-s − 4·8-s + 82·9-s − 44·11-s + 14·12-s − 58·13-s + 56·14-s − 51·16-s − 4·17-s + 164·18-s + 258·19-s − 392·21-s − 88·22-s − 8·23-s + 56·24-s − 116·26-s − 250·27-s − 28·28-s − 396·29-s − 56·31-s − 154·32-s + 616·33-s − 8·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 2.69·3-s − 1/8·4-s − 1.90·6-s + 1.51·7-s − 0.176·8-s + 3.03·9-s − 1.20·11-s + 0.336·12-s − 1.23·13-s + 1.06·14-s − 0.796·16-s − 0.0570·17-s + 2.14·18-s + 3.11·19-s − 4.07·21-s − 0.852·22-s − 0.0725·23-s + 0.476·24-s − 0.874·26-s − 1.78·27-s − 0.188·28-s − 2.53·29-s − 0.324·31-s − 0.850·32-s + 3.24·33-s − 0.0403·34-s + ⋯ |
Λ(s)=(=((58⋅74⋅114)s/2ΓC(s)4L(s)Λ(4−s)
Λ(s)=(=((58⋅74⋅114)s/2ΓC(s+3/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
58⋅74⋅114
|
Sign: |
1
|
Analytic conductor: |
1.66412×108 |
Root analytic conductor: |
10.6573 |
Motivic weight: |
3 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
4
|
Selberg data: |
(8, 58⋅74⋅114, ( :3/2,3/2,3/2,3/2), 1)
|
Particular Values
L(2) |
= |
0 |
L(21) |
= |
0 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 5 | | 1 |
| 7 | C1 | (1−pT)4 |
| 11 | C1 | (1+pT)4 |
good | 2 | C2≀S4 | 1−pT+5T2−p3T3+p6T4−p6T5+5p6T6−p10T7+p12T8 |
| 3 | C2≀S4 | 1+14T+38pT2+698T3+3682T4+698p3T5+38p7T6+14p9T7+p12T8 |
| 13 | C2≀S4 | 1+58T+3862T2+39850T3+2368354T4+39850p3T5+3862p6T6+58p9T7+p12T8 |
| 17 | C2≀S4 | 1+4T+13466T2−198500T3+81336754T4−198500p3T5+13466p6T6+4p9T7+p12T8 |
| 19 | C2≀S4 | 1−258T+2296pT2−5147154T3+489918366T4−5147154p3T5+2296p7T6−258p9T7+p12T8 |
| 23 | C2≀S4 | 1+8T+26276T2−1258456T3+325878550T4−1258456p3T5+26276p6T6+8p9T7+p12T8 |
| 29 | C2≀S4 | 1+396T+138428T2+30598020T3+5584930806T4+30598020p3T5+138428p6T6+396p9T7+p12T8 |
| 31 | C2≀S4 | 1+56T+87274T2+3513992T3+3413701858T4+3513992p3T5+87274p6T6+56p9T7+p12T8 |
| 37 | C2≀S4 | 1+84T+139096T2+12117036T3+8970963870T4+12117036p3T5+139096p6T6+84p9T7+p12T8 |
| 41 | C2≀S4 | 1−52T+191978T2+4000964T3+16303189330T4+4000964p3T5+191978p6T6−52p9T7+p12T8 |
| 43 | C2≀S4 | 1+408T+227224T2+65489736T3+24699468414T4+65489736p3T5+227224p6T6+408p9T7+p12T8 |
| 47 | C2≀S4 | 1+8T+293642T2+14787896T3+39096224482T4+14787896p3T5+293642p6T6+8p9T7+p12T8 |
| 53 | C2≀S4 | 1+624T+731456T2+291124560T3+173869150638T4+291124560p3T5+731456p6T6+624p9T7+p12T8 |
| 59 | C2≀S4 | 1+238T+320090T2−3740918T3+64718281666T4−3740918p3T5+320090p6T6+238p9T7+p12T8 |
| 61 | C2≀S4 | 1+162T+320206T2−56573262T3+48989538114T4−56573262p3T5+320206p6T6+162p9T7+p12T8 |
| 67 | C2≀S4 | 1+20pT+1030948T2+550745180T3+298359795814T4+550745180p3T5+1030948p6T6+20p10T7+p12T8 |
| 71 | C2≀S4 | 1−1788T+2193428T2−1809946572T3+1240921367286T4−1809946572p3T5+2193428p6T6−1788p9T7+p12T8 |
| 73 | C2≀S4 | 1+1456T+1167226T2+565357096T3+283420832722T4+565357096p3T5+1167226p6T6+1456p9T7+p12T8 |
| 79 | C2≀S4 | 1+1324T+2043976T2+1611726940T3+1469807561518T4+1611726940p3T5+2043976p6T6+1324p9T7+p12T8 |
| 83 | C2≀S4 | 1+450T+1903832T2+631295250T3+1545243120894T4+631295250p3T5+1903832p6T6+450p9T7+p12T8 |
| 89 | C2≀S4 | 1+3072T+5688284T2+7201250304T3+6916861622502T4+7201250304p3T5+5688284p6T6+3072p9T7+p12T8 |
| 97 | C2≀S4 | 1−652T+1754332T2−1157057716T3+2404953539782T4−1157057716p3T5+1754332p6T6−652p9T7+p12T8 |
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L(s)=p∏ j=1∏8(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−6.56054503816477307642447452244, −6.27541295474257218050246558202, −5.87227488915221248872508984693, −5.70466952102789921708955412297, −5.60357201226020354468648835936, −5.43650652225122065767353265206, −5.42177629686587462295808339678, −5.21927231854748104596508944415, −5.11328117649201450051913517960, −4.76458300511517225038525932865, −4.48739823661913007015850299937, −4.36955910148303407806303780605, −4.32882645216477692558027505337, −3.96348087064170576557546987920, −3.57638184949549429285793217821, −3.24534135518849802573170881736, −2.95993286772760325130379001315, −2.87902316241498795960511413372, −2.73513262057433906995558452778, −2.05698249644274196921216776357, −1.74991222174700817016838783528, −1.73281192446457696294183096003, −1.48085365700395338844793837556, −0.904184601321824200233560170188, −0.896430905870101369595148155590, 0, 0, 0, 0,
0.896430905870101369595148155590, 0.904184601321824200233560170188, 1.48085365700395338844793837556, 1.73281192446457696294183096003, 1.74991222174700817016838783528, 2.05698249644274196921216776357, 2.73513262057433906995558452778, 2.87902316241498795960511413372, 2.95993286772760325130379001315, 3.24534135518849802573170881736, 3.57638184949549429285793217821, 3.96348087064170576557546987920, 4.32882645216477692558027505337, 4.36955910148303407806303780605, 4.48739823661913007015850299937, 4.76458300511517225038525932865, 5.11328117649201450051913517960, 5.21927231854748104596508944415, 5.42177629686587462295808339678, 5.43650652225122065767353265206, 5.60357201226020354468648835936, 5.70466952102789921708955412297, 5.87227488915221248872508984693, 6.27541295474257218050246558202, 6.56054503816477307642447452244