# Properties

 Degree $32$ Conductor $6.158\times 10^{28}$ Sign $1$ Motivic weight $3$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 16·4-s + 56·7-s + 145·16-s − 612·19-s + 490·25-s − 896·28-s + 1.12e3·31-s − 1.19e3·37-s + 328·43-s + 1.96e3·49-s − 1.63e3·61-s + 452·64-s + 308·67-s + 4.06e3·73-s + 9.79e3·76-s − 2.17e3·79-s − 7.84e3·100-s + 1.38e3·103-s − 1.07e3·109-s + 8.12e3·112-s − 3.20e3·121-s − 1.80e4·124-s + 127-s + 131-s − 3.42e4·133-s + 137-s + 139-s + ⋯
 L(s)  = 1 − 2·4-s + 3.02·7-s + 2.26·16-s − 7.38·19-s + 3.91·25-s − 6.04·28-s + 6.53·31-s − 5.31·37-s + 1.16·43-s + 40/7·49-s − 3.42·61-s + 0.882·64-s + 0.561·67-s + 6.52·73-s + 14.7·76-s − 3.09·79-s − 7.83·100-s + 1.32·103-s − 0.945·109-s + 6.85·112-s − 2.40·121-s − 13.0·124-s + 0.000698·127-s + 0.000666·131-s − 22.3·133-s + 0.000623·137-s + 0.000610·139-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$32$$ Conductor: $$3^{32} \cdot 7^{16}$$ Sign: $1$ Motivic weight: $$3$$ Character: induced by $\chi_{63} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(32,\ 3^{32} \cdot 7^{16} ,\ ( \ : [3/2]^{16} ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.0395433$$ $$L(\frac12)$$ $$\approx$$ $$0.0395433$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$( 1 - 4 p T + 4 p^{2} T^{2} + 4 p^{4} T^{3} - 5359 p^{2} T^{4} + 4 p^{7} T^{5} + 4 p^{8} T^{6} - 4 p^{10} T^{7} + p^{12} T^{8} )^{2}$$
good2 $$1 + p^{4} T^{2} + 111 T^{4} - 249 p^{2} T^{6} - 22563 T^{8} - 99699 p T^{10} - 3667 p^{6} T^{12} + 1401253 p^{3} T^{14} + 9152001 p^{4} T^{16} + 1401253 p^{9} T^{18} - 3667 p^{18} T^{20} - 99699 p^{19} T^{22} - 22563 p^{24} T^{24} - 249 p^{32} T^{26} + 111 p^{36} T^{28} + p^{46} T^{30} + p^{48} T^{32}$$
5 $$1 - 98 p T^{2} + 4047 p^{2} T^{4} - 13467734 T^{6} + 2004738389 T^{8} - 76183349844 p T^{10} + 60059957637514 T^{12} - 6902983281820528 T^{14} + 756156244566152106 T^{16} - 6902983281820528 p^{6} T^{18} + 60059957637514 p^{12} T^{20} - 76183349844 p^{19} T^{22} + 2004738389 p^{24} T^{24} - 13467734 p^{30} T^{26} + 4047 p^{38} T^{28} - 98 p^{43} T^{30} + p^{48} T^{32}$$
11 $$1 + 3202 T^{2} + 1797447 T^{4} - 5608813218 T^{6} - 4531249774539 T^{8} + 15353827063546164 T^{10} + 27034229520225124106 T^{12} +$$$$48\!\cdots\!88$$$$T^{14} -$$$$31\!\cdots\!02$$$$T^{16} +$$$$48\!\cdots\!88$$$$p^{6} T^{18} + 27034229520225124106 p^{12} T^{20} + 15353827063546164 p^{18} T^{22} - 4531249774539 p^{24} T^{24} - 5608813218 p^{30} T^{26} + 1797447 p^{36} T^{28} + 3202 p^{42} T^{30} + p^{48} T^{32}$$
13 $$( 1 - 4502 T^{2} + 10097461 T^{4} - 24107307194 T^{6} + 60089639013832 T^{8} - 24107307194 p^{6} T^{10} + 10097461 p^{12} T^{12} - 4502 p^{18} T^{14} + p^{24} T^{16} )^{2}$$
17 $$1 - 16948 T^{2} + 84203208 T^{4} - 399083701352 T^{6} + 6508712666429186 T^{8} - 42615253832024962764 T^{10} +$$$$11\!\cdots\!04$$$$T^{12} -$$$$92\!\cdots\!84$$$$T^{14} +$$$$77\!\cdots\!75$$$$T^{16} -$$$$92\!\cdots\!84$$$$p^{6} T^{18} +$$$$11\!\cdots\!04$$$$p^{12} T^{20} - 42615253832024962764 p^{18} T^{22} + 6508712666429186 p^{24} T^{24} - 399083701352 p^{30} T^{26} + 84203208 p^{36} T^{28} - 16948 p^{42} T^{30} + p^{48} T^{32}$$
19 $$( 1 + 306 T + 57757 T^{2} + 8122770 T^{3} + 990235135 T^{4} + 107235980892 T^{5} + 10750308055972 T^{6} + 990048157019604 T^{7} + 85251569497003126 T^{8} + 990048157019604 p^{3} T^{9} + 10750308055972 p^{6} T^{10} + 107235980892 p^{9} T^{11} + 990235135 p^{12} T^{12} + 8122770 p^{15} T^{13} + 57757 p^{18} T^{14} + 306 p^{21} T^{15} + p^{24} T^{16} )^{2}$$
23 $$1 + 54964 T^{2} + 1326151704 T^{4} + 25415665738344 T^{6} + 524305919802157986 T^{8} +$$$$89\!\cdots\!72$$$$T^{10} +$$$$12\!\cdots\!96$$$$T^{12} +$$$$17\!\cdots\!32$$$$T^{14} +$$$$23\!\cdots\!87$$$$T^{16} +$$$$17\!\cdots\!32$$$$p^{6} T^{18} +$$$$12\!\cdots\!96$$$$p^{12} T^{20} +$$$$89\!\cdots\!72$$$$p^{18} T^{22} + 524305919802157986 p^{24} T^{24} + 25415665738344 p^{30} T^{26} + 1326151704 p^{36} T^{28} + 54964 p^{42} T^{30} + p^{48} T^{32}$$
29 $$( 1 - 102034 T^{2} + 6213617737 T^{4} - 245008733294114 T^{6} + 7064972052959541764 T^{8} - 245008733294114 p^{6} T^{10} + 6213617737 p^{12} T^{12} - 102034 p^{18} T^{14} + p^{24} T^{16} )^{2}$$
31 $$( 1 - 564 T + 243388 T^{2} - 77468784 T^{3} + 21469133071 T^{4} - 5073861272220 T^{5} + 1094289736794676 T^{6} - 211880592093523344 T^{7} + 38213365513366414528 T^{8} - 211880592093523344 p^{3} T^{9} + 1094289736794676 p^{6} T^{10} - 5073861272220 p^{9} T^{11} + 21469133071 p^{12} T^{12} - 77468784 p^{15} T^{13} + 243388 p^{18} T^{14} - 564 p^{21} T^{15} + p^{24} T^{16} )^{2}$$
37 $$( 1 + 598 T + 47601 T^{2} - 11729418 T^{3} + 10699876539 T^{4} + 3891656263464 T^{5} + 114219217143080 T^{6} + 66547288562775176 T^{7} + 48227467333244423670 T^{8} + 66547288562775176 p^{3} T^{9} + 114219217143080 p^{6} T^{10} + 3891656263464 p^{9} T^{11} + 10699876539 p^{12} T^{12} - 11729418 p^{15} T^{13} + 47601 p^{18} T^{14} + 598 p^{21} T^{15} + p^{24} T^{16} )^{2}$$
41 $$( 1 + 148396 T^{2} + 19955386648 T^{4} + 1828971842691940 T^{6} +$$$$14\!\cdots\!50$$$$T^{8} + 1828971842691940 p^{6} T^{10} + 19955386648 p^{12} T^{12} + 148396 p^{18} T^{14} + p^{24} T^{16} )^{2}$$
43 $$( 1 - 82 T + 122239 T^{2} - 47448038 T^{3} + 6939760430 T^{4} - 47448038 p^{3} T^{5} + 122239 p^{6} T^{6} - 82 p^{9} T^{7} + p^{12} T^{8} )^{4}$$
47 $$1 - 358312 T^{2} + 90535322508 T^{4} - 16795335458693168 T^{6} +$$$$25\!\cdots\!46$$$$T^{8} -$$$$71\!\cdots\!68$$$$p T^{10} +$$$$39\!\cdots\!64$$$$T^{12} -$$$$42\!\cdots\!76$$$$T^{14} +$$$$45\!\cdots\!75$$$$T^{16} -$$$$42\!\cdots\!76$$$$p^{6} T^{18} +$$$$39\!\cdots\!64$$$$p^{12} T^{20} -$$$$71\!\cdots\!68$$$$p^{19} T^{22} +$$$$25\!\cdots\!46$$$$p^{24} T^{24} - 16795335458693168 p^{30} T^{26} + 90535322508 p^{36} T^{28} - 358312 p^{42} T^{30} + p^{48} T^{32}$$
53 $$1 + 506674 T^{2} + 85866141819 T^{4} + 10208516436240318 T^{6} +$$$$32\!\cdots\!49$$$$T^{8} +$$$$63\!\cdots\!00$$$$T^{10} +$$$$61\!\cdots\!74$$$$T^{12} +$$$$12\!\cdots\!24$$$$T^{14} +$$$$28\!\cdots\!38$$$$T^{16} +$$$$12\!\cdots\!24$$$$p^{6} T^{18} +$$$$61\!\cdots\!74$$$$p^{12} T^{20} +$$$$63\!\cdots\!00$$$$p^{18} T^{22} +$$$$32\!\cdots\!49$$$$p^{24} T^{24} + 10208516436240318 p^{30} T^{26} + 85866141819 p^{36} T^{28} + 506674 p^{42} T^{30} + p^{48} T^{32}$$
59 $$1 - 399442 T^{2} + 89994325083 T^{4} - 32591407040941742 T^{6} +$$$$78\!\cdots\!45$$$$T^{8} -$$$$13\!\cdots\!56$$$$T^{10} +$$$$43\!\cdots\!26$$$$T^{12} -$$$$98\!\cdots\!64$$$$T^{14} +$$$$16\!\cdots\!22$$$$T^{16} -$$$$98\!\cdots\!64$$$$p^{6} T^{18} +$$$$43\!\cdots\!26$$$$p^{12} T^{20} -$$$$13\!\cdots\!56$$$$p^{18} T^{22} +$$$$78\!\cdots\!45$$$$p^{24} T^{24} - 32591407040941742 p^{30} T^{26} + 89994325083 p^{36} T^{28} - 399442 p^{42} T^{30} + p^{48} T^{32}$$
61 $$( 1 + 816 T + 837478 T^{2} + 502269216 T^{3} + 333755818642 T^{4} + 188353178559720 T^{5} + 93910648157586304 T^{6} + 50370689927417233344 T^{7} +$$$$21\!\cdots\!87$$$$T^{8} + 50370689927417233344 p^{3} T^{9} + 93910648157586304 p^{6} T^{10} + 188353178559720 p^{9} T^{11} + 333755818642 p^{12} T^{12} + 502269216 p^{15} T^{13} + 837478 p^{18} T^{14} + 816 p^{21} T^{15} + p^{24} T^{16} )^{2}$$
67 $$( 1 - 154 T - 1009305 T^{2} + 101871994 T^{3} + 602011276205 T^{4} - 36961847361936 T^{5} - 261850611517253414 T^{6} + 66134569555683020 p T^{7} +$$$$89\!\cdots\!66$$$$T^{8} + 66134569555683020 p^{4} T^{9} - 261850611517253414 p^{6} T^{10} - 36961847361936 p^{9} T^{11} + 602011276205 p^{12} T^{12} + 101871994 p^{15} T^{13} - 1009305 p^{18} T^{14} - 154 p^{21} T^{15} + p^{24} T^{16} )^{2}$$
71 $$( 1 - 2546056 T^{2} + 2933736697204 T^{4} - 1997708473027518488 T^{6} +$$$$87\!\cdots\!74$$$$T^{8} - 1997708473027518488 p^{6} T^{10} + 2933736697204 p^{12} T^{12} - 2546056 p^{18} T^{14} + p^{24} T^{16} )^{2}$$
73 $$( 1 - 2034 T + 2450047 T^{2} - 2178403830 T^{3} + 1347031106821 T^{4} - 430919574332148 T^{5} - 113107388808572534 T^{6} +$$$$30\!\cdots\!80$$$$T^{7} -$$$$25\!\cdots\!10$$$$T^{8} +$$$$30\!\cdots\!80$$$$p^{3} T^{9} - 113107388808572534 p^{6} T^{10} - 430919574332148 p^{9} T^{11} + 1347031106821 p^{12} T^{12} - 2178403830 p^{15} T^{13} + 2450047 p^{18} T^{14} - 2034 p^{21} T^{15} + p^{24} T^{16} )^{2}$$
79 $$( 1 + 1088 T - 269028 T^{2} - 1109682680 T^{3} - 391267378201 T^{4} + 358084200932532 T^{5} + 285753619209423700 T^{6} - 15705523238748286852 T^{7} -$$$$10\!\cdots\!68$$$$T^{8} - 15705523238748286852 p^{3} T^{9} + 285753619209423700 p^{6} T^{10} + 358084200932532 p^{9} T^{11} - 391267378201 p^{12} T^{12} - 1109682680 p^{15} T^{13} - 269028 p^{18} T^{14} + 1088 p^{21} T^{15} + p^{24} T^{16} )^{2}$$
83 $$( 1 + 2632486 T^{2} + 3636170382349 T^{4} + 3366834686358200962 T^{6} +$$$$22\!\cdots\!84$$$$T^{8} + 3366834686358200962 p^{6} T^{10} + 3636170382349 p^{12} T^{12} + 2632486 p^{18} T^{14} + p^{24} T^{16} )^{2}$$
89 $$1 - 2548864 T^{2} + 4462766350596 T^{4} - 5306850100326352640 T^{6} +$$$$51\!\cdots\!26$$$$T^{8} -$$$$40\!\cdots\!68$$$$T^{10} +$$$$27\!\cdots\!80$$$$T^{12} -$$$$17\!\cdots\!04$$$$T^{14} +$$$$11\!\cdots\!27$$$$T^{16} -$$$$17\!\cdots\!04$$$$p^{6} T^{18} +$$$$27\!\cdots\!80$$$$p^{12} T^{20} -$$$$40\!\cdots\!68$$$$p^{18} T^{22} +$$$$51\!\cdots\!26$$$$p^{24} T^{24} - 5306850100326352640 p^{30} T^{26} + 4462766350596 p^{36} T^{28} - 2548864 p^{42} T^{30} + p^{48} T^{32}$$
97 $$( 1 - 2978570 T^{2} + 5898756336049 T^{4} - 7901916867899068322 T^{6} +$$$$83\!\cdots\!24$$$$T^{8} - 7901916867899068322 p^{6} T^{10} + 5898756336049 p^{12} T^{12} - 2978570 p^{18} T^{14} + p^{24} T^{16} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$