Properties

Degree 32
Conductor $ 2^{80} \cdot 7^{16} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 32·9-s + 48·17-s − 92·25-s + 112·29-s + 8·37-s − 64·49-s − 24·53-s − 360·61-s + 72·73-s + 538·81-s + 408·89-s − 672·101-s − 336·109-s − 1.04e3·113-s + 640·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 1.53e3·153-s + 157-s + 163-s + 167-s + 384·169-s + 173-s + ⋯
L(s)  = 1  − 3.55·9-s + 2.82·17-s − 3.67·25-s + 3.86·29-s + 8/37·37-s − 1.30·49-s − 0.452·53-s − 5.90·61-s + 0.986·73-s + 6.64·81-s + 4.58·89-s − 6.65·101-s − 3.08·109-s − 9.20·113-s + 5.28·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s − 10.0·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2.27·169-s + 0.00578·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(32\)
\( N \)  =  \(2^{80} \cdot 7^{16}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{224} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((32,\ 2^{80} \cdot 7^{16} ,\ ( \ : [1]^{16} ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.73153\)
\(L(\frac12)\)  \(\approx\)  \(2.73153\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7\}$,\(F_p(T)\) is a polynomial of degree 32. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 31.
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + 64 T^{2} + 5084 T^{4} + 338880 T^{6} + 309158 p^{2} T^{8} + 338880 p^{4} T^{10} + 5084 p^{8} T^{12} + 64 p^{12} T^{14} + p^{16} T^{16} \)
good3 \( 1 + 32 T^{2} + 2 p^{5} T^{4} + 3776 T^{6} + 6841 T^{8} - 54784 p T^{10} - 57374 p^{3} T^{12} - 63136 p^{3} T^{14} + 657172 p^{4} T^{16} - 63136 p^{7} T^{18} - 57374 p^{11} T^{20} - 54784 p^{13} T^{22} + 6841 p^{16} T^{24} + 3776 p^{20} T^{26} + 2 p^{29} T^{28} + 32 p^{28} T^{30} + p^{32} T^{32} \)
5 \( ( 1 + 46 T^{2} + 1073 T^{4} - 1008 p T^{5} + 9678 T^{6} - 50064 p T^{7} - 145276 T^{8} - 50064 p^{3} T^{9} + 9678 p^{4} T^{10} - 1008 p^{7} T^{11} + 1073 p^{8} T^{12} + 46 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
11 \( 1 - 640 T^{2} + 227030 T^{4} - 52241408 T^{6} + 8348702825 T^{8} - 853428477056 T^{10} + 32116185305766 T^{12} + 6409181104515840 T^{14} - 1320087233250529292 T^{16} + 6409181104515840 p^{4} T^{18} + 32116185305766 p^{8} T^{20} - 853428477056 p^{12} T^{22} + 8348702825 p^{16} T^{24} - 52241408 p^{20} T^{26} + 227030 p^{24} T^{28} - 640 p^{28} T^{30} + p^{32} T^{32} \)
13 \( ( 1 - 192 T^{2} - 16036 T^{4} + 1118400 T^{6} + 5937174 p^{2} T^{8} + 1118400 p^{4} T^{10} - 16036 p^{8} T^{12} - 192 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
17 \( ( 1 - 24 T + 766 T^{2} - 13776 T^{3} + 249905 T^{4} - 2407680 T^{5} - 7157346 T^{6} + 486721512 T^{7} - 15232969756 T^{8} + 486721512 p^{2} T^{9} - 7157346 p^{4} T^{10} - 2407680 p^{6} T^{11} + 249905 p^{8} T^{12} - 13776 p^{10} T^{13} + 766 p^{12} T^{14} - 24 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
19 \( 1 + 528 T^{2} + 24262 T^{4} + 11189472 T^{6} + 10328689 p^{2} T^{8} - 6420105050688 T^{10} + 940099701135094 T^{12} + 1121910404491838544 T^{14} + \)\(16\!\cdots\!80\)\( T^{16} + 1121910404491838544 p^{4} T^{18} + 940099701135094 p^{8} T^{20} - 6420105050688 p^{12} T^{22} + 10328689 p^{18} T^{24} + 11189472 p^{20} T^{26} + 24262 p^{24} T^{28} + 528 p^{28} T^{30} + p^{32} T^{32} \)
23 \( 1 - 928 T^{2} - 110362 T^{4} + 76027072 T^{6} + 161836431161 T^{8} - 2090789193920 T^{10} - 54941991557559978 T^{12} + 14303404978544854944 T^{14} - \)\(12\!\cdots\!04\)\( T^{16} + 14303404978544854944 p^{4} T^{18} - 54941991557559978 p^{8} T^{20} - 2090789193920 p^{12} T^{22} + 161836431161 p^{16} T^{24} + 76027072 p^{20} T^{26} - 110362 p^{24} T^{28} - 928 p^{28} T^{30} + p^{32} T^{32} \)
29 \( ( 1 - 28 T + 1376 T^{2} - 1428 p T^{3} + 2055086 T^{4} - 1428 p^{3} T^{5} + 1376 p^{4} T^{6} - 28 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
31 \( 1 + 4480 T^{2} + 9494774 T^{4} + 14306099840 T^{6} + 619917826775 p T^{8} + 24003699927942080 T^{10} + 26963891194383292806 T^{12} + \)\(88\!\cdots\!40\)\( p T^{14} + \)\(26\!\cdots\!04\)\( T^{16} + \)\(88\!\cdots\!40\)\( p^{5} T^{18} + 26963891194383292806 p^{8} T^{20} + 24003699927942080 p^{12} T^{22} + 619917826775 p^{17} T^{24} + 14306099840 p^{20} T^{26} + 9494774 p^{24} T^{28} + 4480 p^{28} T^{30} + p^{32} T^{32} \)
37 \( ( 1 - 4 T - 2770 T^{2} + 121912 T^{3} + 3850313 T^{4} - 247260872 T^{5} + 4348488942 T^{6} + 233812310196 T^{7} - 10747661313644 T^{8} + 233812310196 p^{2} T^{9} + 4348488942 p^{4} T^{10} - 247260872 p^{6} T^{11} + 3850313 p^{8} T^{12} + 121912 p^{10} T^{13} - 2770 p^{12} T^{14} - 4 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
41 \( ( 1 - 4384 T^{2} + 14370524 T^{4} - 32104130784 T^{6} + 62557592438726 T^{8} - 32104130784 p^{4} T^{10} + 14370524 p^{8} T^{12} - 4384 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
43 \( ( 1 + 11176 T^{2} + 58293116 T^{4} + 188186249880 T^{6} + 415246366681670 T^{8} + 188186249880 p^{4} T^{10} + 58293116 p^{8} T^{12} + 11176 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
47 \( 1 + 7504 T^{2} + 21182918 T^{4} + 37880838368 T^{6} + 100768600793177 T^{8} + 218689095748226816 T^{10} + \)\(22\!\cdots\!70\)\( T^{12} + \)\(76\!\cdots\!12\)\( T^{14} + \)\(29\!\cdots\!24\)\( T^{16} + \)\(76\!\cdots\!12\)\( p^{4} T^{18} + \)\(22\!\cdots\!70\)\( p^{8} T^{20} + 218689095748226816 p^{12} T^{22} + 100768600793177 p^{16} T^{24} + 37880838368 p^{20} T^{26} + 21182918 p^{24} T^{28} + 7504 p^{28} T^{30} + p^{32} T^{32} \)
53 \( ( 1 + 12 T - 5314 T^{2} - 123432 T^{3} + 13289689 T^{4} + 412137720 T^{5} + 8001350174 T^{6} - 662204794044 T^{7} - 79868347316108 T^{8} - 662204794044 p^{2} T^{9} + 8001350174 p^{4} T^{10} + 412137720 p^{6} T^{11} + 13289689 p^{8} T^{12} - 123432 p^{10} T^{13} - 5314 p^{12} T^{14} + 12 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
59 \( 1 + 12496 T^{2} + 69591254 T^{4} + 231196127072 T^{6} + 548940902299433 T^{8} + 1224313485336173888 T^{10} + \)\(76\!\cdots\!62\)\( T^{12} - \)\(17\!\cdots\!52\)\( T^{14} - \)\(10\!\cdots\!32\)\( T^{16} - \)\(17\!\cdots\!52\)\( p^{4} T^{18} + \)\(76\!\cdots\!62\)\( p^{8} T^{20} + 1224313485336173888 p^{12} T^{22} + 548940902299433 p^{16} T^{24} + 231196127072 p^{20} T^{26} + 69591254 p^{24} T^{28} + 12496 p^{28} T^{30} + p^{32} T^{32} \)
61 \( ( 1 + 180 T + 27726 T^{2} + 3046680 T^{3} + 308297129 T^{4} + 26423205000 T^{5} + 2082233555790 T^{6} + 145145432104140 T^{7} + 9348584465944212 T^{8} + 145145432104140 p^{2} T^{9} + 2082233555790 p^{4} T^{10} + 26423205000 p^{6} T^{11} + 308297129 p^{8} T^{12} + 3046680 p^{10} T^{13} + 27726 p^{12} T^{14} + 180 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
67 \( 1 - 5792 T^{2} - 24227898 T^{4} + 351601426496 T^{6} - 205859799182695 T^{8} - 8707473271916577024 T^{10} + \)\(34\!\cdots\!66\)\( T^{12} + \)\(81\!\cdots\!24\)\( T^{14} - \)\(92\!\cdots\!56\)\( T^{16} + \)\(81\!\cdots\!24\)\( p^{4} T^{18} + \)\(34\!\cdots\!66\)\( p^{8} T^{20} - 8707473271916577024 p^{12} T^{22} - 205859799182695 p^{16} T^{24} + 351601426496 p^{20} T^{26} - 24227898 p^{24} T^{28} - 5792 p^{28} T^{30} + p^{32} T^{32} \)
71 \( ( 1 + 27624 T^{2} + 370534588 T^{4} + 3151580461272 T^{6} + 18747759294515334 T^{8} + 3151580461272 p^{4} T^{10} + 370534588 p^{8} T^{12} + 27624 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
73 \( ( 1 - 36 T + 13374 T^{2} - 465912 T^{3} + 80365289 T^{4} - 3011641896 T^{5} + 480881406846 T^{6} - 17666763587772 T^{7} + 3025613230783092 T^{8} - 17666763587772 p^{2} T^{9} + 480881406846 p^{4} T^{10} - 3011641896 p^{6} T^{11} + 80365289 p^{8} T^{12} - 465912 p^{10} T^{13} + 13374 p^{12} T^{14} - 36 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
79 \( 1 - 128 p T^{2} - 55382202 T^{4} + 565038672512 T^{6} + 4616239476368153 T^{8} - 23803283016229531968 T^{10} - \)\(25\!\cdots\!02\)\( T^{12} + \)\(18\!\cdots\!72\)\( T^{14} + \)\(13\!\cdots\!08\)\( T^{16} + \)\(18\!\cdots\!72\)\( p^{4} T^{18} - \)\(25\!\cdots\!02\)\( p^{8} T^{20} - 23803283016229531968 p^{12} T^{22} + 4616239476368153 p^{16} T^{24} + 565038672512 p^{20} T^{26} - 55382202 p^{24} T^{28} - 128 p^{29} T^{30} + p^{32} T^{32} \)
83 \( ( 1 - 24232 T^{2} + 235912988 T^{4} - 1035732354840 T^{6} + 3398417338650758 T^{8} - 1035732354840 p^{4} T^{10} + 235912988 p^{8} T^{12} - 24232 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
89 \( ( 1 - 204 T + 42982 T^{2} - 5938440 T^{3} + 800796785 T^{4} - 88063687656 T^{5} + 106175287590 p T^{6} - 894257389252164 T^{7} + 83094368751630116 T^{8} - 894257389252164 p^{2} T^{9} + 106175287590 p^{5} T^{10} - 88063687656 p^{6} T^{11} + 800796785 p^{8} T^{12} - 5938440 p^{10} T^{13} + 42982 p^{12} T^{14} - 204 p^{14} T^{15} + p^{16} T^{16} )^{2} \)
97 \( ( 1 - 51968 T^{2} + 13461148 p T^{4} - 20766356196608 T^{6} + 230652504659485894 T^{8} - 20766356196608 p^{4} T^{10} + 13461148 p^{9} T^{12} - 51968 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.24396323745900055749980818137, −3.09159643606012623314266550482, −2.98691146840457627844815349859, −2.90514753192835319204292775890, −2.86923057644656756638241113287, −2.76887458825470767910026283811, −2.72252672323508552798108079157, −2.69501997856623799487442414189, −2.62677627598154578256915258673, −2.31429007654766115830924356662, −2.20808274445534665079774031857, −2.05407864261556472736598683598, −2.05237330897291087060328208001, −1.92630429750291607928702576293, −1.68727972118057331974029908246, −1.63971955801756715894419809430, −1.49587634292256155199659513530, −1.19061468779408055357411222707, −1.17782187939062965665763266610, −1.16372000153567695624490027420, −0.971338917017686934594998859805, −0.56300404280826388858502741493, −0.31997725516089513036062134426, −0.30971118560324316011289760649, −0.24036364074616699664341298168, 0.24036364074616699664341298168, 0.30971118560324316011289760649, 0.31997725516089513036062134426, 0.56300404280826388858502741493, 0.971338917017686934594998859805, 1.16372000153567695624490027420, 1.17782187939062965665763266610, 1.19061468779408055357411222707, 1.49587634292256155199659513530, 1.63971955801756715894419809430, 1.68727972118057331974029908246, 1.92630429750291607928702576293, 2.05237330897291087060328208001, 2.05407864261556472736598683598, 2.20808274445534665079774031857, 2.31429007654766115830924356662, 2.62677627598154578256915258673, 2.69501997856623799487442414189, 2.72252672323508552798108079157, 2.76887458825470767910026283811, 2.86923057644656756638241113287, 2.90514753192835319204292775890, 2.98691146840457627844815349859, 3.09159643606012623314266550482, 3.24396323745900055749980818137

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.