Dirichlet series
| L(s) = 1 | − 8·2-s + 12·3-s + 24·4-s + 40·5-s − 96·6-s + 19·7-s + 54·9-s − 320·10-s − 21·11-s + 288·12-s + 15·13-s − 152·14-s + 480·15-s − 240·16-s − 8·17-s − 432·18-s − 90·19-s + 960·20-s + 228·21-s + 168·22-s − 145·23-s + 900·25-s − 120·26-s + 456·28-s − 245·29-s − 3.84e3·30-s + 768·32-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 2.30·3-s + 3·4-s + 3.57·5-s − 6.53·6-s + 1.02·7-s + 2·9-s − 10.1·10-s − 0.575·11-s + 6.92·12-s + 0.320·13-s − 2.90·14-s + 8.26·15-s − 3.75·16-s − 0.114·17-s − 5.65·18-s − 1.08·19-s + 10.7·20-s + 2.36·21-s + 1.62·22-s − 1.31·23-s + 36/5·25-s − 0.905·26-s + 3.07·28-s − 1.56·29-s − 23.3·30-s + 4.24·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.86041\times 10^{10}\) |
| Root analytic conductor: | \(4.79695\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(5.280797897\) |
| \(L(\frac12)\) | \(\approx\) | \(5.280797897\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p T + p^{2} T^{2} )^{4} \) |
| 3 | \( ( 1 - p T + p^{2} T^{2} )^{4} \) | |
| 5 | \( ( 1 - p T )^{8} \) | |
| 13 | \( 1 - 15 T + 167 p T^{2} - 3630 p T^{3} + 58278 p^{2} T^{4} - 3630 p^{4} T^{5} + 167 p^{7} T^{6} - 15 p^{9} T^{7} + p^{12} T^{8} \) | |
| good | 7 | \( 1 - 19 T - 304 T^{2} + 13857 T^{3} - 53531 T^{4} - 4076852 T^{5} + 83290974 T^{6} + 556368920 T^{7} - 37338874236 T^{8} + 556368920 p^{3} T^{9} + 83290974 p^{6} T^{10} - 4076852 p^{9} T^{11} - 53531 p^{12} T^{12} + 13857 p^{15} T^{13} - 304 p^{18} T^{14} - 19 p^{21} T^{15} + p^{24} T^{16} \) |
| 11 | \( 1 + 21 T - 201 p T^{2} + 55218 T^{3} + 4095689 T^{4} - 122218425 T^{5} + 146633250 p T^{6} + 147200533641 T^{7} - 4919173405698 T^{8} + 147200533641 p^{3} T^{9} + 146633250 p^{7} T^{10} - 122218425 p^{9} T^{11} + 4095689 p^{12} T^{12} + 55218 p^{15} T^{13} - 201 p^{19} T^{14} + 21 p^{21} T^{15} + p^{24} T^{16} \) | |
| 17 | \( 1 + 8 T - 12196 T^{2} + 574716 T^{3} + 84478974 T^{4} - 5176397734 T^{5} - 172417672524 T^{6} + 15883002845116 T^{7} + 113514059860143 T^{8} + 15883002845116 p^{3} T^{9} - 172417672524 p^{6} T^{10} - 5176397734 p^{9} T^{11} + 84478974 p^{12} T^{12} + 574716 p^{15} T^{13} - 12196 p^{18} T^{14} + 8 p^{21} T^{15} + p^{24} T^{16} \) | |
| 19 | \( 1 + 90 T - 15280 T^{2} - 1135140 T^{3} + 166599542 T^{4} + 6811380690 T^{5} - 1653513749280 T^{6} - 12020725386630 T^{7} + 14179570550487503 T^{8} - 12020725386630 p^{3} T^{9} - 1653513749280 p^{6} T^{10} + 6811380690 p^{9} T^{11} + 166599542 p^{12} T^{12} - 1135140 p^{15} T^{13} - 15280 p^{18} T^{14} + 90 p^{21} T^{15} + p^{24} T^{16} \) | |
| 23 | \( 1 + 145 T + 8069 T^{2} + 2143442 T^{3} + 330162565 T^{4} + 1659734709 p T^{5} + 258309083546 p T^{6} + 1134148179577 p^{2} T^{7} + 93082376041106 p^{2} T^{8} + 1134148179577 p^{5} T^{9} + 258309083546 p^{7} T^{10} + 1659734709 p^{10} T^{11} + 330162565 p^{12} T^{12} + 2143442 p^{15} T^{13} + 8069 p^{18} T^{14} + 145 p^{21} T^{15} + p^{24} T^{16} \) | |
| 29 | \( 1 + 245 T - 3029 T^{2} + 3209432 T^{3} + 1233094561 T^{4} - 74196597243 T^{5} - 2511353574226 T^{6} + 1531326300055347 T^{7} - 344238764472094816 T^{8} + 1531326300055347 p^{3} T^{9} - 2511353574226 p^{6} T^{10} - 74196597243 p^{9} T^{11} + 1233094561 p^{12} T^{12} + 3209432 p^{15} T^{13} - 3029 p^{18} T^{14} + 245 p^{21} T^{15} + p^{24} T^{16} \) | |
| 31 | \( ( 1 + 71678 T^{2} - 3362652 T^{3} + 2452927971 T^{4} - 3362652 p^{3} T^{5} + 71678 p^{6} T^{6} + p^{12} T^{8} )^{2} \) | |
| 37 | \( 1 - 55 T - 118345 T^{2} - 2451468 T^{3} + 7565467783 T^{4} + 412093333177 T^{5} - 234162449978814 T^{6} - 13194810129906007 T^{7} + 5871641665491432420 T^{8} - 13194810129906007 p^{3} T^{9} - 234162449978814 p^{6} T^{10} + 412093333177 p^{9} T^{11} + 7565467783 p^{12} T^{12} - 2451468 p^{15} T^{13} - 118345 p^{18} T^{14} - 55 p^{21} T^{15} + p^{24} T^{16} \) | |
| 41 | \( 1 - 216 T - 153438 T^{2} + 600092 p T^{3} + 13979244662 T^{4} - 1030033669258 T^{5} - 1202319031437620 T^{6} + 24038033854920076 T^{7} + 89480318621803804507 T^{8} + 24038033854920076 p^{3} T^{9} - 1202319031437620 p^{6} T^{10} - 1030033669258 p^{9} T^{11} + 13979244662 p^{12} T^{12} + 600092 p^{16} T^{13} - 153438 p^{18} T^{14} - 216 p^{21} T^{15} + p^{24} T^{16} \) | |
| 43 | \( 1 - 80 T - 54780 T^{2} - 1799596 T^{3} - 663428367 T^{4} + 467521231946 T^{5} + 504022874183864 T^{6} - 62719772309701014 T^{7} - 39036727199688218380 T^{8} - 62719772309701014 p^{3} T^{9} + 504022874183864 p^{6} T^{10} + 467521231946 p^{9} T^{11} - 663428367 p^{12} T^{12} - 1799596 p^{15} T^{13} - 54780 p^{18} T^{14} - 80 p^{21} T^{15} + p^{24} T^{16} \) | |
| 47 | \( ( 1 - 41 T + 375970 T^{2} - 14000491 T^{3} + 56569368236 T^{4} - 14000491 p^{3} T^{5} + 375970 p^{6} T^{6} - 41 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 53 | \( ( 1 - 458 T + 464812 T^{2} - 166176150 T^{3} + 94202582726 T^{4} - 166176150 p^{3} T^{5} + 464812 p^{6} T^{6} - 458 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 59 | \( 1 + 155 T - 782773 T^{2} - 62555916 T^{3} + 389264957247 T^{4} + 18340387647611 T^{5} - 125499035904028806 T^{6} - 1288312633087766141 T^{7} + \)\(30\!\cdots\!58\)\( T^{8} - 1288312633087766141 p^{3} T^{9} - 125499035904028806 p^{6} T^{10} + 18340387647611 p^{9} T^{11} + 389264957247 p^{12} T^{12} - 62555916 p^{15} T^{13} - 782773 p^{18} T^{14} + 155 p^{21} T^{15} + p^{24} T^{16} \) | |
| 61 | \( 1 - 693 T + 51180 T^{2} + 220790211 T^{3} - 171575971447 T^{4} + 74172111374616 T^{5} - 5000430295910562 T^{6} - 18594695893638161442 T^{7} + \)\(14\!\cdots\!36\)\( T^{8} - 18594695893638161442 p^{3} T^{9} - 5000430295910562 p^{6} T^{10} + 74172111374616 p^{9} T^{11} - 171575971447 p^{12} T^{12} + 220790211 p^{15} T^{13} + 51180 p^{18} T^{14} - 693 p^{21} T^{15} + p^{24} T^{16} \) | |
| 67 | \( 1 + 837 T + 549992 T^{2} - 106600155 T^{3} - 234954410779 T^{4} - 215795156751336 T^{5} - 12643035426176250 T^{6} + 42279277484230337484 T^{7} + \)\(49\!\cdots\!28\)\( T^{8} + 42279277484230337484 p^{3} T^{9} - 12643035426176250 p^{6} T^{10} - 215795156751336 p^{9} T^{11} - 234954410779 p^{12} T^{12} - 106600155 p^{15} T^{13} + 549992 p^{18} T^{14} + 837 p^{21} T^{15} + p^{24} T^{16} \) | |
| 71 | \( 1 + 1158 T - 358230 T^{2} - 460072500 T^{3} + 525113555498 T^{4} + 281836234736400 T^{5} - 219542513093028384 T^{6} - 10254243619458308298 T^{7} + \)\(12\!\cdots\!35\)\( T^{8} - 10254243619458308298 p^{3} T^{9} - 219542513093028384 p^{6} T^{10} + 281836234736400 p^{9} T^{11} + 525113555498 p^{12} T^{12} - 460072500 p^{15} T^{13} - 358230 p^{18} T^{14} + 1158 p^{21} T^{15} + p^{24} T^{16} \) | |
| 73 | \( ( 1 - 885 T + 1444677 T^{2} - 1027749660 T^{3} + 821884659670 T^{4} - 1027749660 p^{3} T^{5} + 1444677 p^{6} T^{6} - 885 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 79 | \( ( 1 - 298 T + 881132 T^{2} - 287245436 T^{3} + 682993239729 T^{4} - 287245436 p^{3} T^{5} + 881132 p^{6} T^{6} - 298 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 83 | \( ( 1 + 2328 T + 4265076 T^{2} + 4715932008 T^{3} + 4318634082934 T^{4} + 4715932008 p^{3} T^{5} + 4265076 p^{6} T^{6} + 2328 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 89 | \( 1 - 830 T - 1810138 T^{2} + 812427156 T^{3} + 2559557076942 T^{4} - 494835594353516 T^{5} - 2494225882477214976 T^{6} + \)\(17\!\cdots\!26\)\( T^{7} + \)\(18\!\cdots\!63\)\( T^{8} + \)\(17\!\cdots\!26\)\( p^{3} T^{9} - 2494225882477214976 p^{6} T^{10} - 494835594353516 p^{9} T^{11} + 2559557076942 p^{12} T^{12} + 812427156 p^{15} T^{13} - 1810138 p^{18} T^{14} - 830 p^{21} T^{15} + p^{24} T^{16} \) | |
| 97 | \( 1 + 1491 T - 633358 T^{2} - 2749963935 T^{3} - 803500390987 T^{4} + 2063173403921052 T^{5} + 1594816710281352066 T^{6} - \)\(49\!\cdots\!68\)\( T^{7} - \)\(13\!\cdots\!72\)\( T^{8} - \)\(49\!\cdots\!68\)\( p^{3} T^{9} + 1594816710281352066 p^{6} T^{10} + 2063173403921052 p^{9} T^{11} - 803500390987 p^{12} T^{12} - 2749963935 p^{15} T^{13} - 633358 p^{18} T^{14} + 1491 p^{21} T^{15} + p^{24} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.55026435783784683658394292319, −4.33171915074924446736141136111, −4.18672726562177936072242070170, −4.13791734543422848752938401106, −4.03773581639094021797201768604, −3.56085157997953341728128257331, −3.53219170734941430450739885966, −3.33497058459241256190144300406, −3.25827267625829487147032748784, −2.99371081874416039832013490632, −2.58208283030915595213101907084, −2.53893823859874676795435034424, −2.49685814893832979347688981622, −2.45972570352068104915238782589, −2.08604216642669767713231337950, −2.00002758051655339103459108170, −1.90860388857025835564529192171, −1.66511764168655626427649321229, −1.60345371790369465463636194799, −1.48824726439733924887391282118, −1.12524190094835488891566117490, −0.790761046335738072961336185440, −0.63970358369222478092408819581, −0.60556055060180375748426768788, −0.16364354490017366994004996655, 0.16364354490017366994004996655, 0.60556055060180375748426768788, 0.63970358369222478092408819581, 0.790761046335738072961336185440, 1.12524190094835488891566117490, 1.48824726439733924887391282118, 1.60345371790369465463636194799, 1.66511764168655626427649321229, 1.90860388857025835564529192171, 2.00002758051655339103459108170, 2.08604216642669767713231337950, 2.45972570352068104915238782589, 2.49685814893832979347688981622, 2.53893823859874676795435034424, 2.58208283030915595213101907084, 2.99371081874416039832013490632, 3.25827267625829487147032748784, 3.33497058459241256190144300406, 3.53219170734941430450739885966, 3.56085157997953341728128257331, 4.03773581639094021797201768604, 4.13791734543422848752938401106, 4.18672726562177936072242070170, 4.33171915074924446736141136111, 4.55026435783784683658394292319
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.