Properties

Label 32-33e16-1.1-c5e16-0-0
Degree $32$
Conductor $1.978\times 10^{24}$
Sign $1$
Analytic cond. $3.79133\times 10^{11}$
Root an. cond. $2.30057$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 54·3-s − 98·4-s + 1.34e3·9-s + 5.29e3·12-s + 2.69e3·16-s + 2.43e4·25-s − 2.13e4·27-s − 1.19e4·31-s − 1.32e5·36-s + 9.35e3·37-s − 1.45e5·48-s + 1.33e5·49-s + 4.16e4·64-s − 3.64e5·67-s − 1.31e6·75-s + 1.94e5·81-s + 6.46e5·93-s + 1.19e5·97-s − 2.38e6·100-s + 3.32e4·103-s + 2.09e6·108-s − 5.05e5·111-s + 5.23e5·121-s + 1.17e6·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 3.46·3-s − 3.06·4-s + 5.54·9-s + 10.6·12-s + 2.63·16-s + 7.78·25-s − 5.63·27-s − 2.23·31-s − 16.9·36-s + 1.12·37-s − 9.11·48-s + 7.96·49-s + 1.27·64-s − 9.90·67-s − 26.9·75-s + 3.28·81-s + 7.75·93-s + 1.29·97-s − 23.8·100-s + 0.309·103-s + 17.2·108-s − 3.89·111-s + 3.24·121-s + 6.85·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(3^{16} \cdot 11^{16}\)
Sign: $1$
Analytic conductor: \(3.79133\times 10^{11}\)
Root analytic conductor: \(2.30057\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{33} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 3^{16} \cdot 11^{16} ,\ ( \ : [5/2]^{16} ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(0.3014260601\)
\(L(\frac12)\) \(\approx\) \(0.3014260601\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + p^{3} T + 140 p T^{2} + 197 p^{3} T^{3} + 370 p^{5} T^{4} + 197 p^{8} T^{5} + 140 p^{11} T^{6} + p^{18} T^{7} + p^{20} T^{8} )^{2} \)
11 \( 1 - 523232 T^{2} + 1199686204 p^{2} T^{4} - 1900207683104 p^{4} T^{6} + 21115726569958 p^{8} T^{8} - 1900207683104 p^{14} T^{10} + 1199686204 p^{22} T^{12} - 523232 p^{30} T^{14} + p^{40} T^{16} \)
good2 \( ( 1 + 49 T^{2} + 1127 p T^{4} + 2351 p^{5} T^{6} + 100019 p^{5} T^{8} + 2351 p^{15} T^{10} + 1127 p^{21} T^{12} + 49 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
5 \( ( 1 - 12161 T^{2} + 77169586 T^{4} - 342209368091 T^{6} + 1188160998850786 T^{8} - 342209368091 p^{10} T^{10} + 77169586 p^{20} T^{12} - 12161 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
7 \( ( 1 - 66944 T^{2} + 2582701168 T^{4} - 68803513304192 T^{6} + 1331111347651038046 T^{8} - 68803513304192 p^{10} T^{10} + 2582701168 p^{20} T^{12} - 66944 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
13 \( ( 1 - 516428 T^{2} + 556537683592 T^{4} - 204833489102832932 T^{6} + \)\(11\!\cdots\!86\)\( T^{8} - 204833489102832932 p^{10} T^{10} + 556537683592 p^{20} T^{12} - 516428 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
17 \( ( 1 + 8439940 T^{2} + 33982555335940 T^{4} + 85217776986692055292 T^{6} + \)\(14\!\cdots\!66\)\( T^{8} + 85217776986692055292 p^{10} T^{10} + 33982555335940 p^{20} T^{12} + 8439940 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
19 \( ( 1 - 13543964 T^{2} + 90505658469700 T^{4} - \)\(38\!\cdots\!32\)\( T^{6} + \)\(59\!\cdots\!14\)\( p T^{8} - \)\(38\!\cdots\!32\)\( p^{10} T^{10} + 90505658469700 p^{20} T^{12} - 13543964 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
23 \( ( 1 - 33097349 T^{2} + 537570467487082 T^{4} - \)\(56\!\cdots\!55\)\( T^{6} + \)\(42\!\cdots\!34\)\( T^{8} - \)\(56\!\cdots\!55\)\( p^{10} T^{10} + 537570467487082 p^{20} T^{12} - 33097349 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
29 \( ( 1 + 123527836 T^{2} + 7350528812010628 T^{4} + \)\(27\!\cdots\!72\)\( T^{6} + \)\(66\!\cdots\!86\)\( T^{8} + \)\(27\!\cdots\!72\)\( p^{10} T^{10} + 7350528812010628 p^{20} T^{12} + 123527836 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
31 \( ( 1 + 2995 T + 70788940 T^{2} + 114310683799 T^{3} + 2385311558674918 T^{4} + 114310683799 p^{5} T^{5} + 70788940 p^{10} T^{6} + 2995 p^{15} T^{7} + p^{20} T^{8} )^{4} \)
37 \( ( 1 - 2339 T + 164657482 T^{2} - 9201379805 p T^{3} + 14946695959559434 T^{4} - 9201379805 p^{6} T^{5} + 164657482 p^{10} T^{6} - 2339 p^{15} T^{7} + p^{20} T^{8} )^{4} \)
41 \( ( 1 + 188324764 T^{2} + 32097757212980164 T^{4} + \)\(48\!\cdots\!32\)\( T^{6} + \)\(71\!\cdots\!02\)\( T^{8} + \)\(48\!\cdots\!32\)\( p^{10} T^{10} + 32097757212980164 p^{20} T^{12} + 188324764 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
43 \( ( 1 - 2939492 p T^{2} - 4682972857463132 T^{4} - \)\(14\!\cdots\!28\)\( T^{6} + \)\(88\!\cdots\!86\)\( T^{8} - \)\(14\!\cdots\!28\)\( p^{10} T^{10} - 4682972857463132 p^{20} T^{12} - 2939492 p^{31} T^{14} + p^{40} T^{16} )^{2} \)
47 \( ( 1 - 1205991188 T^{2} + 710569059038402200 T^{4} - \)\(26\!\cdots\!04\)\( T^{6} + \)\(72\!\cdots\!18\)\( T^{8} - \)\(26\!\cdots\!04\)\( p^{10} T^{10} + 710569059038402200 p^{20} T^{12} - 1205991188 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
53 \( ( 1 - 2266729916 T^{2} + 47798623138710152 p T^{4} - \)\(18\!\cdots\!28\)\( T^{6} + \)\(89\!\cdots\!66\)\( T^{8} - \)\(18\!\cdots\!28\)\( p^{10} T^{10} + 47798623138710152 p^{21} T^{12} - 2266729916 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
59 \( ( 1 - 2925095969 T^{2} + 4804505895375021442 T^{4} - \)\(54\!\cdots\!07\)\( T^{6} + \)\(44\!\cdots\!26\)\( T^{8} - \)\(54\!\cdots\!07\)\( p^{10} T^{10} + 4804505895375021442 p^{20} T^{12} - 2925095969 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
61 \( ( 1 - 4448729132 T^{2} + 9755875885631452168 T^{4} - \)\(13\!\cdots\!76\)\( T^{6} + \)\(13\!\cdots\!78\)\( T^{8} - \)\(13\!\cdots\!76\)\( p^{10} T^{10} + 9755875885631452168 p^{20} T^{12} - 4448729132 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
67 \( ( 1 + 91033 T + 7075541680 T^{2} + 333861885426913 T^{3} + 14553622094902605454 T^{4} + 333861885426913 p^{5} T^{5} + 7075541680 p^{10} T^{6} + 91033 p^{15} T^{7} + p^{20} T^{8} )^{4} \)
71 \( ( 1 - 6293085533 T^{2} + 24199856209130825914 T^{4} - \)\(65\!\cdots\!11\)\( T^{6} + \)\(13\!\cdots\!58\)\( T^{8} - \)\(65\!\cdots\!11\)\( p^{10} T^{10} + 24199856209130825914 p^{20} T^{12} - 6293085533 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
73 \( ( 1 - 5123388680 T^{2} + 12857066040387608668 T^{4} - \)\(37\!\cdots\!32\)\( T^{6} + \)\(10\!\cdots\!46\)\( T^{8} - \)\(37\!\cdots\!32\)\( p^{10} T^{10} + 12857066040387608668 p^{20} T^{12} - 5123388680 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
79 \( ( 1 - 12358523072 T^{2} + 48649054043753229424 T^{4} - \)\(10\!\cdots\!64\)\( T^{6} - \)\(31\!\cdots\!82\)\( T^{8} - \)\(10\!\cdots\!64\)\( p^{10} T^{10} + 48649054043753229424 p^{20} T^{12} - 12358523072 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
83 \( ( 1 + 26359778920 T^{2} + \)\(32\!\cdots\!08\)\( T^{4} + \)\(23\!\cdots\!64\)\( T^{6} + \)\(11\!\cdots\!38\)\( T^{8} + \)\(23\!\cdots\!64\)\( p^{10} T^{10} + \)\(32\!\cdots\!08\)\( p^{20} T^{12} + 26359778920 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
89 \( ( 1 - 33267956981 T^{2} + \)\(52\!\cdots\!78\)\( T^{4} - \)\(51\!\cdots\!87\)\( T^{6} + \)\(34\!\cdots\!06\)\( T^{8} - \)\(51\!\cdots\!87\)\( p^{10} T^{10} + \)\(52\!\cdots\!78\)\( p^{20} T^{12} - 33267956981 p^{30} T^{14} + p^{40} T^{16} )^{2} \)
97 \( ( 1 - 29963 T + 16625594254 T^{2} - 436974456024497 T^{3} + \)\(21\!\cdots\!50\)\( T^{4} - 436974456024497 p^{5} T^{5} + 16625594254 p^{10} T^{6} - 29963 p^{15} T^{7} + p^{20} T^{8} )^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.38451809607134296854856953657, −4.34579097973186112987807189854, −4.34062773245689249281738671893, −4.28620382905283889518747395942, −3.88573204932314259153819131485, −3.67154108912693610473334433957, −3.62450022703965031050779871180, −3.25960849837752797013653804291, −3.22105433788878974359730528812, −2.95622490059246783513194538887, −2.91695891253145075028212819410, −2.80871458074465628853742241098, −2.66828938336921261610340364704, −2.46700860963799403565050440010, −2.05634583321655603632335110553, −1.79621340541335969187927488482, −1.79249125903142106547943534661, −1.38088066271545675011886398154, −1.01164343239908765584285203637, −0.908526356593065660483960061786, −0.864255537177516523426622375664, −0.841618810941280350611269483588, −0.41839507253354455158098168910, −0.30186416634300794197142816645, −0.17793631367946754520878291401, 0.17793631367946754520878291401, 0.30186416634300794197142816645, 0.41839507253354455158098168910, 0.841618810941280350611269483588, 0.864255537177516523426622375664, 0.908526356593065660483960061786, 1.01164343239908765584285203637, 1.38088066271545675011886398154, 1.79249125903142106547943534661, 1.79621340541335969187927488482, 2.05634583321655603632335110553, 2.46700860963799403565050440010, 2.66828938336921261610340364704, 2.80871458074465628853742241098, 2.91695891253145075028212819410, 2.95622490059246783513194538887, 3.22105433788878974359730528812, 3.25960849837752797013653804291, 3.62450022703965031050779871180, 3.67154108912693610473334433957, 3.88573204932314259153819131485, 4.28620382905283889518747395942, 4.34062773245689249281738671893, 4.34579097973186112987807189854, 4.38451809607134296854856953657

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

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