# Properties

 Degree 32 Conductor $2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}$ Sign $1$ Motivic weight 2 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 16·2-s + 128·4-s − 16·5-s − 672·8-s + 256·10-s + 8·11-s − 32·13-s + 2.54e3·16-s + 56·17-s − 2.04e3·20-s − 128·22-s + 24·23-s + 148·25-s + 512·26-s − 112·31-s − 7.10e3·32-s − 896·34-s − 152·37-s + 1.07e4·40-s + 1.02e3·44-s − 384·46-s + 80·47-s − 2.36e3·50-s − 4.09e3·52-s + 48·53-s − 128·55-s + 96·61-s + ⋯
 L(s)  = 1 − 8·2-s + 32·4-s − 3.19·5-s − 84·8-s + 25.5·10-s + 8/11·11-s − 2.46·13-s + 159·16-s + 3.29·17-s − 102.·20-s − 5.81·22-s + 1.04·23-s + 5.91·25-s + 19.6·26-s − 3.61·31-s − 222·32-s − 26.3·34-s − 4.10·37-s + 268.·40-s + 23.2·44-s − 8.34·46-s + 1.70·47-s − 47.3·50-s − 78.7·52-s + 0.905·53-s − 2.32·55-s + 1.57·61-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$32$$ $$N$$ = $$2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}$$ $$\varepsilon$$ = $1$ motivic weight = $$2$$ character : induced by $\chi_{210} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(32,\ 2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [1]^{16} ),\ 1 )$$ $$L(\frac{3}{2})$$ $$\approx$$ $$0.00500243$$ $$L(\frac12)$$ $$\approx$$ $$0.00500243$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 32. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 31.
$p$$F_p(T)$
bad2 $$( 1 + p T + p T^{2} )^{8}$$
3 $$( 1 + p^{2} T^{4} )^{4}$$
5 $$1 + 16 T + 108 T^{2} + 48 p T^{3} - 1736 T^{4} - 15856 T^{5} - 26268 T^{6} + 70704 p T^{7} + 117534 p^{2} T^{8} + 70704 p^{3} T^{9} - 26268 p^{4} T^{10} - 15856 p^{6} T^{11} - 1736 p^{8} T^{12} + 48 p^{11} T^{13} + 108 p^{12} T^{14} + 16 p^{14} T^{15} + p^{16} T^{16}$$
7 $$( 1 + p^{2} T^{4} )^{4}$$
good11 $$( 1 - 4 T + 428 T^{2} - 3020 T^{3} + 112132 T^{4} - 750180 T^{5} + 21255892 T^{6} - 12021156 p T^{7} + 2897491254 T^{8} - 12021156 p^{3} T^{9} + 21255892 p^{4} T^{10} - 750180 p^{6} T^{11} + 112132 p^{8} T^{12} - 3020 p^{10} T^{13} + 428 p^{12} T^{14} - 4 p^{14} T^{15} + p^{16} T^{16} )^{2}$$
13 $$1 + 32 T + 512 T^{2} + 12624 T^{3} + 210400 T^{4} + 1218608 T^{5} + 10953344 T^{6} - 12361888 T^{7} - 9595251012 T^{8} - 144909422048 T^{9} - 1442470310272 T^{10} - 30698489798640 T^{11} - 15354999406944 p T^{12} + 2586969858854768 T^{13} + 29532641854346496 T^{14} + 799237423444942944 T^{15} + 19007129532539001606 T^{16} + 799237423444942944 p^{2} T^{17} + 29532641854346496 p^{4} T^{18} + 2586969858854768 p^{6} T^{19} - 15354999406944 p^{9} T^{20} - 30698489798640 p^{10} T^{21} - 1442470310272 p^{12} T^{22} - 144909422048 p^{14} T^{23} - 9595251012 p^{16} T^{24} - 12361888 p^{18} T^{25} + 10953344 p^{20} T^{26} + 1218608 p^{22} T^{27} + 210400 p^{24} T^{28} + 12624 p^{26} T^{29} + 512 p^{28} T^{30} + 32 p^{30} T^{31} + p^{32} T^{32}$$
17 $$1 - 56 T + 1568 T^{2} - 29832 T^{3} + 267808 T^{4} + 199880 p T^{5} - 165234592 T^{6} + 3465012472 T^{7} - 48572233284 T^{8} + 491667950696 T^{9} - 4716183819232 T^{10} + 39445096533336 T^{11} + 280624802792928 T^{12} - 26208629063564312 T^{13} + 761979769352889696 T^{14} - 15987258252632453928 T^{15} +$$$$28\!\cdots\!74$$$$T^{16} - 15987258252632453928 p^{2} T^{17} + 761979769352889696 p^{4} T^{18} - 26208629063564312 p^{6} T^{19} + 280624802792928 p^{8} T^{20} + 39445096533336 p^{10} T^{21} - 4716183819232 p^{12} T^{22} + 491667950696 p^{14} T^{23} - 48572233284 p^{16} T^{24} + 3465012472 p^{18} T^{25} - 165234592 p^{20} T^{26} + 199880 p^{23} T^{27} + 267808 p^{24} T^{28} - 29832 p^{26} T^{29} + 1568 p^{28} T^{30} - 56 p^{30} T^{31} + p^{32} T^{32}$$
19 $$1 - 2056 T^{2} + 2230944 T^{4} - 1603740248 T^{6} + 43114975316 p T^{8} - 297045269538504 T^{10} + 69987022118810464 T^{12} - 7175141044809308632 T^{14} -$$$$43\!\cdots\!46$$$$T^{16} - 7175141044809308632 p^{4} T^{18} + 69987022118810464 p^{8} T^{20} - 297045269538504 p^{12} T^{22} + 43114975316 p^{17} T^{24} - 1603740248 p^{20} T^{26} + 2230944 p^{24} T^{28} - 2056 p^{28} T^{30} + p^{32} T^{32}$$
23 $$1 - 24 T + 288 T^{2} - 17192 T^{3} + 221472 T^{4} - 2391768 T^{5} + 141400928 T^{6} - 3172207336 T^{7} + 137674628156 T^{8} - 274022502200 T^{9} - 6993233578720 T^{10} - 103772124341000 T^{11} - 60508914798085408 T^{12} + 933592691352233992 T^{13} - 15427273581773601184 T^{14} +$$$$57\!\cdots\!36$$$$T^{15} -$$$$64\!\cdots\!26$$$$T^{16} +$$$$57\!\cdots\!36$$$$p^{2} T^{17} - 15427273581773601184 p^{4} T^{18} + 933592691352233992 p^{6} T^{19} - 60508914798085408 p^{8} T^{20} - 103772124341000 p^{10} T^{21} - 6993233578720 p^{12} T^{22} - 274022502200 p^{14} T^{23} + 137674628156 p^{16} T^{24} - 3172207336 p^{18} T^{25} + 141400928 p^{20} T^{26} - 2391768 p^{22} T^{27} + 221472 p^{24} T^{28} - 17192 p^{26} T^{29} + 288 p^{28} T^{30} - 24 p^{30} T^{31} + p^{32} T^{32}$$
29 $$1 - 4032 T^{2} + 8105976 T^{4} - 11130637632 T^{6} + 11901856158364 T^{8} - 11208853722734016 T^{10} + 10516466734892486856 T^{12} -$$$$99\!\cdots\!40$$$$T^{14} +$$$$87\!\cdots\!46$$$$T^{16} -$$$$99\!\cdots\!40$$$$p^{4} T^{18} + 10516466734892486856 p^{8} T^{20} - 11208853722734016 p^{12} T^{22} + 11901856158364 p^{16} T^{24} - 11130637632 p^{20} T^{26} + 8105976 p^{24} T^{28} - 4032 p^{28} T^{30} + p^{32} T^{32}$$
31 $$( 1 + 56 T + 4484 T^{2} + 186776 T^{3} + 8992264 T^{4} + 304202984 T^{5} + 11800345036 T^{6} + 348556649544 T^{7} + 12266333612430 T^{8} + 348556649544 p^{2} T^{9} + 11800345036 p^{4} T^{10} + 304202984 p^{6} T^{11} + 8992264 p^{8} T^{12} + 186776 p^{10} T^{13} + 4484 p^{12} T^{14} + 56 p^{14} T^{15} + p^{16} T^{16} )^{2}$$
37 $$1 + 152 T + 11552 T^{2} + 598344 T^{3} + 25962168 T^{4} + 1167158904 T^{5} + 56500959840 T^{6} + 2639490056104 T^{7} + 111109511071516 T^{8} + 4242418231122872 T^{9} + 154876079427922976 T^{10} + 5762441784032775208 T^{11} +$$$$22\!\cdots\!72$$$$T^{12} +$$$$89\!\cdots\!08$$$$T^{13} +$$$$32\!\cdots\!12$$$$T^{14} +$$$$10\!\cdots\!40$$$$T^{15} +$$$$37\!\cdots\!46$$$$T^{16} +$$$$10\!\cdots\!40$$$$p^{2} T^{17} +$$$$32\!\cdots\!12$$$$p^{4} T^{18} +$$$$89\!\cdots\!08$$$$p^{6} T^{19} +$$$$22\!\cdots\!72$$$$p^{8} T^{20} + 5762441784032775208 p^{10} T^{21} + 154876079427922976 p^{12} T^{22} + 4242418231122872 p^{14} T^{23} + 111109511071516 p^{16} T^{24} + 2639490056104 p^{18} T^{25} + 56500959840 p^{20} T^{26} + 1167158904 p^{22} T^{27} + 25962168 p^{24} T^{28} + 598344 p^{26} T^{29} + 11552 p^{28} T^{30} + 152 p^{30} T^{31} + p^{32} T^{32}$$
41 $$( 1 + 8252 T^{2} - 19936 T^{3} + 34068728 T^{4} - 127251488 T^{5} + 92291784532 T^{6} - 367862933760 T^{7} + 180432835211182 T^{8} - 367862933760 p^{2} T^{9} + 92291784532 p^{4} T^{10} - 127251488 p^{6} T^{11} + 34068728 p^{8} T^{12} - 19936 p^{10} T^{13} + 8252 p^{12} T^{14} + p^{16} T^{16} )^{2}$$
43 $$1 - 158592 T^{3} + 2725896 T^{4} - 61004160 T^{5} + 12575711232 T^{6} - 618250768128 T^{7} + 18017190018460 T^{8} - 968512496926464 T^{9} + 65630298643144704 T^{10} - 3092793950726971008 T^{11} +$$$$10\!\cdots\!80$$$$T^{12} -$$$$37\!\cdots\!52$$$$T^{13} +$$$$33\!\cdots\!64$$$$T^{14} -$$$$12\!\cdots\!48$$$$T^{15} +$$$$20\!\cdots\!26$$$$T^{16} -$$$$12\!\cdots\!48$$$$p^{2} T^{17} +$$$$33\!\cdots\!64$$$$p^{4} T^{18} -$$$$37\!\cdots\!52$$$$p^{6} T^{19} +$$$$10\!\cdots\!80$$$$p^{8} T^{20} - 3092793950726971008 p^{10} T^{21} + 65630298643144704 p^{12} T^{22} - 968512496926464 p^{14} T^{23} + 18017190018460 p^{16} T^{24} - 618250768128 p^{18} T^{25} + 12575711232 p^{20} T^{26} - 61004160 p^{22} T^{27} + 2725896 p^{24} T^{28} - 158592 p^{26} T^{29} + p^{32} T^{32}$$
47 $$1 - 80 T + 3200 T^{2} - 185424 T^{3} + 10961416 T^{4} - 416599088 T^{5} + 15442425728 T^{6} - 918151881776 T^{7} + 40326873674268 T^{8} - 452413353633424 T^{9} - 5098985333045632 T^{10} + 1563817770500267376 T^{11} -$$$$31\!\cdots\!76$$$$T^{12} +$$$$17\!\cdots\!60$$$$T^{13} -$$$$66\!\cdots\!16$$$$T^{14} +$$$$36\!\cdots\!52$$$$T^{15} -$$$$19\!\cdots\!38$$$$T^{16} +$$$$36\!\cdots\!52$$$$p^{2} T^{17} -$$$$66\!\cdots\!16$$$$p^{4} T^{18} +$$$$17\!\cdots\!60$$$$p^{6} T^{19} -$$$$31\!\cdots\!76$$$$p^{8} T^{20} + 1563817770500267376 p^{10} T^{21} - 5098985333045632 p^{12} T^{22} - 452413353633424 p^{14} T^{23} + 40326873674268 p^{16} T^{24} - 918151881776 p^{18} T^{25} + 15442425728 p^{20} T^{26} - 416599088 p^{22} T^{27} + 10961416 p^{24} T^{28} - 185424 p^{26} T^{29} + 3200 p^{28} T^{30} - 80 p^{30} T^{31} + p^{32} T^{32}$$
53 $$1 - 48 T + 1152 T^{2} - 172480 T^{3} + 14460640 T^{4} - 33746752 T^{5} - 164137984 T^{6} + 7272556080 T^{7} - 144052676687940 T^{8} + 6754246786116880 T^{9} - 156566570729557248 T^{10} + 21067810577807322688 T^{11} -$$$$28\!\cdots\!60$$$$T^{12} -$$$$50\!\cdots\!00$$$$T^{13} +$$$$11\!\cdots\!80$$$$T^{14} -$$$$15\!\cdots\!88$$$$T^{15} +$$$$20\!\cdots\!78$$$$T^{16} -$$$$15\!\cdots\!88$$$$p^{2} T^{17} +$$$$11\!\cdots\!80$$$$p^{4} T^{18} -$$$$50\!\cdots\!00$$$$p^{6} T^{19} -$$$$28\!\cdots\!60$$$$p^{8} T^{20} + 21067810577807322688 p^{10} T^{21} - 156566570729557248 p^{12} T^{22} + 6754246786116880 p^{14} T^{23} - 144052676687940 p^{16} T^{24} + 7272556080 p^{18} T^{25} - 164137984 p^{20} T^{26} - 33746752 p^{22} T^{27} + 14460640 p^{24} T^{28} - 172480 p^{26} T^{29} + 1152 p^{28} T^{30} - 48 p^{30} T^{31} + p^{32} T^{32}$$
59 $$1 - 14320 T^{2} + 157767608 T^{4} - 1244890527696 T^{6} + 8207935216533276 T^{8} - 45520264315988706288 T^{10} +$$$$21\!\cdots\!36$$$$T^{12} -$$$$91\!\cdots\!96$$$$T^{14} +$$$$33\!\cdots\!38$$$$T^{16} -$$$$91\!\cdots\!96$$$$p^{4} T^{18} +$$$$21\!\cdots\!36$$$$p^{8} T^{20} - 45520264315988706288 p^{12} T^{22} + 8207935216533276 p^{16} T^{24} - 1244890527696 p^{20} T^{26} + 157767608 p^{24} T^{28} - 14320 p^{28} T^{30} + p^{32} T^{32}$$
61 $$( 1 - 48 T + 22640 T^{2} - 673936 T^{3} + 215932124 T^{4} - 3175993520 T^{5} + 1217381291920 T^{6} - 6110146782096 T^{7} + 5029652273881990 T^{8} - 6110146782096 p^{2} T^{9} + 1217381291920 p^{4} T^{10} - 3175993520 p^{6} T^{11} + 215932124 p^{8} T^{12} - 673936 p^{10} T^{13} + 22640 p^{12} T^{14} - 48 p^{14} T^{15} + p^{16} T^{16} )^{2}$$
67 $$1 + 80 T + 3200 T^{2} - 9104 p T^{3} - 25803832 T^{4} + 2674348464 T^{5} + 482550620032 T^{6} + 12521071726512 T^{7} - 1529742891382500 T^{8} - 116256625005179504 T^{9} + 3984919157383710336 T^{10} +$$$$89\!\cdots\!92$$$$T^{11} +$$$$25\!\cdots\!88$$$$T^{12} -$$$$21\!\cdots\!76$$$$T^{13} -$$$$18\!\cdots\!68$$$$T^{14} +$$$$71\!\cdots\!08$$$$T^{15} +$$$$10\!\cdots\!46$$$$T^{16} +$$$$71\!\cdots\!08$$$$p^{2} T^{17} -$$$$18\!\cdots\!68$$$$p^{4} T^{18} -$$$$21\!\cdots\!76$$$$p^{6} T^{19} +$$$$25\!\cdots\!88$$$$p^{8} T^{20} +$$$$89\!\cdots\!92$$$$p^{10} T^{21} + 3984919157383710336 p^{12} T^{22} - 116256625005179504 p^{14} T^{23} - 1529742891382500 p^{16} T^{24} + 12521071726512 p^{18} T^{25} + 482550620032 p^{20} T^{26} + 2674348464 p^{22} T^{27} - 25803832 p^{24} T^{28} - 9104 p^{27} T^{29} + 3200 p^{28} T^{30} + 80 p^{30} T^{31} + p^{32} T^{32}$$
71 $$( 1 - 268 T + 44204 T^{2} - 5071044 T^{3} + 471192708 T^{4} - 37316092140 T^{5} + 2755016710036 T^{6} - 195026045020868 T^{7} + 13901563235868662 T^{8} - 195026045020868 p^{2} T^{9} + 2755016710036 p^{4} T^{10} - 37316092140 p^{6} T^{11} + 471192708 p^{8} T^{12} - 5071044 p^{10} T^{13} + 44204 p^{12} T^{14} - 268 p^{14} T^{15} + p^{16} T^{16} )^{2}$$
73 $$1 - 638960 T^{3} - 86919200 T^{4} - 369113360 T^{5} + 204134940800 T^{6} + 33110468899200 T^{7} + 5206311620706876 T^{8} + 26720496193596800 T^{9} - 3344897125184227200 T^{10} -$$$$13\!\cdots\!20$$$$T^{11} -$$$$22\!\cdots\!00$$$$T^{12} -$$$$53\!\cdots\!20$$$$T^{13} +$$$$75\!\cdots\!00$$$$T^{14} +$$$$36\!\cdots\!00$$$$T^{15} +$$$$78\!\cdots\!66$$$$T^{16} +$$$$36\!\cdots\!00$$$$p^{2} T^{17} +$$$$75\!\cdots\!00$$$$p^{4} T^{18} -$$$$53\!\cdots\!20$$$$p^{6} T^{19} -$$$$22\!\cdots\!00$$$$p^{8} T^{20} -$$$$13\!\cdots\!20$$$$p^{10} T^{21} - 3344897125184227200 p^{12} T^{22} + 26720496193596800 p^{14} T^{23} + 5206311620706876 p^{16} T^{24} + 33110468899200 p^{18} T^{25} + 204134940800 p^{20} T^{26} - 369113360 p^{22} T^{27} - 86919200 p^{24} T^{28} - 638960 p^{26} T^{29} + p^{32} T^{32}$$
79 $$1 - 52208 T^{2} + 1391680568 T^{4} - 25119513250000 T^{6} + 343967562165562012 T^{8} -$$$$37\!\cdots\!72$$$$T^{10} +$$$$34\!\cdots\!24$$$$T^{12} -$$$$27\!\cdots\!40$$$$T^{14} +$$$$18\!\cdots\!30$$$$T^{16} -$$$$27\!\cdots\!40$$$$p^{4} T^{18} +$$$$34\!\cdots\!24$$$$p^{8} T^{20} -$$$$37\!\cdots\!72$$$$p^{12} T^{22} + 343967562165562012 p^{16} T^{24} - 25119513250000 p^{20} T^{26} + 1391680568 p^{24} T^{28} - 52208 p^{28} T^{30} + p^{32} T^{32}$$
83 $$1 + 256 T + 32768 T^{2} + 2489472 T^{3} + 78882184 T^{4} + 642592896 T^{5} + 678427795456 T^{6} + 150782876556800 T^{7} + 17212084280490396 T^{8} + 1275723698299503616 T^{9} + 84640034952293916672 T^{10} +$$$$62\!\cdots\!76$$$$T^{11} +$$$$45\!\cdots\!48$$$$T^{12} +$$$$40\!\cdots\!04$$$$T^{13} +$$$$47\!\cdots\!44$$$$T^{14} +$$$$59\!\cdots\!24$$$$T^{15} +$$$$59\!\cdots\!02$$$$T^{16} +$$$$59\!\cdots\!24$$$$p^{2} T^{17} +$$$$47\!\cdots\!44$$$$p^{4} T^{18} +$$$$40\!\cdots\!04$$$$p^{6} T^{19} +$$$$45\!\cdots\!48$$$$p^{8} T^{20} +$$$$62\!\cdots\!76$$$$p^{10} T^{21} + 84640034952293916672 p^{12} T^{22} + 1275723698299503616 p^{14} T^{23} + 17212084280490396 p^{16} T^{24} + 150782876556800 p^{18} T^{25} + 678427795456 p^{20} T^{26} + 642592896 p^{22} T^{27} + 78882184 p^{24} T^{28} + 2489472 p^{26} T^{29} + 32768 p^{28} T^{30} + 256 p^{30} T^{31} + p^{32} T^{32}$$
89 $$1 - 60568 T^{2} + 1632136448 T^{4} - 25689569431240 T^{6} + 260052896713761532 T^{8} -$$$$17\!\cdots\!32$$$$T^{10} +$$$$70\!\cdots\!24$$$$T^{12} -$$$$68\!\cdots\!20$$$$T^{14} -$$$$92\!\cdots\!90$$$$T^{16} -$$$$68\!\cdots\!20$$$$p^{4} T^{18} +$$$$70\!\cdots\!24$$$$p^{8} T^{20} -$$$$17\!\cdots\!32$$$$p^{12} T^{22} + 260052896713761532 p^{16} T^{24} - 25689569431240 p^{20} T^{26} + 1632136448 p^{24} T^{28} - 60568 p^{28} T^{30} + p^{32} T^{32}$$
97 $$1 - 688 T + 236672 T^{2} - 56991072 T^{3} + 10854023776 T^{4} - 1693042723360 T^{5} + 219961022412800 T^{6} - 24020408470599760 T^{7} + 2158083026478171708 T^{8} -$$$$14\!\cdots\!44$$$$T^{9} +$$$$52\!\cdots\!16$$$$T^{10} +$$$$44\!\cdots\!04$$$$T^{11} -$$$$12\!\cdots\!36$$$$T^{12} +$$$$17\!\cdots\!52$$$$T^{13} -$$$$20\!\cdots\!48$$$$T^{14} +$$$$21\!\cdots\!08$$$$T^{15} -$$$$20\!\cdots\!18$$$$T^{16} +$$$$21\!\cdots\!08$$$$p^{2} T^{17} -$$$$20\!\cdots\!48$$$$p^{4} T^{18} +$$$$17\!\cdots\!52$$$$p^{6} T^{19} -$$$$12\!\cdots\!36$$$$p^{8} T^{20} +$$$$44\!\cdots\!04$$$$p^{10} T^{21} +$$$$52\!\cdots\!16$$$$p^{12} T^{22} -$$$$14\!\cdots\!44$$$$p^{14} T^{23} + 2158083026478171708 p^{16} T^{24} - 24020408470599760 p^{18} T^{25} + 219961022412800 p^{20} T^{26} - 1693042723360 p^{22} T^{27} + 10854023776 p^{24} T^{28} - 56991072 p^{26} T^{29} + 236672 p^{28} T^{30} - 688 p^{30} T^{31} + p^{32} T^{32}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}