Properties

Label 32-300e16-1.1-c8e16-0-1
Degree $32$
Conductor $4.305\times 10^{39}$
Sign $1$
Analytic cond. $2.47691\times 10^{33}$
Root an. cond. $11.0550$
Motivic weight $8$
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  + 7.37e3·9-s + 4.80e4·19-s + 4.70e5·31-s − 3.62e7·49-s + 1.88e7·61-s + 9.07e7·79-s + 3.12e7·81-s − 3.02e8·109-s + 1.48e9·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 6.70e9·169-s + 3.54e8·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 1.12·9-s + 0.368·19-s + 0.509·31-s − 6.28·49-s + 1.36·61-s + 2.33·79-s + 0.726·81-s − 2.14·109-s + 6.93·121-s − 8.21·169-s + 0.414·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+4)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 3^{16} \cdot 5^{32}\)
Sign: $1$
Analytic conductor: \(2.47691\times 10^{33}\)
Root analytic conductor: \(11.0550\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{300} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 3^{16} \cdot 5^{32} ,\ ( \ : [4]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.4933637809\)
\(L(\frac12)\) \(\approx\) \(0.4933637809\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 7378 T^{2} + 95296 p^{5} T^{4} - 1720414 p^{10} T^{6} + 2793715870 p^{12} T^{8} - 1720414 p^{26} T^{10} + 95296 p^{37} T^{12} - 7378 p^{48} T^{14} + p^{64} T^{16} \)
5 \( 1 \)
good7 \( ( 1 + 18104974 T^{2} + 3192552604480 p^{2} T^{4} + 501986334724936786 p^{4} T^{6} + \)\(69\!\cdots\!66\)\( p^{6} T^{8} + 501986334724936786 p^{20} T^{10} + 3192552604480 p^{34} T^{12} + 18104974 p^{48} T^{14} + p^{64} T^{16} )^{2} \)
11 \( ( 1 - 743676888 T^{2} + 302962005747738748 T^{4} - \)\(81\!\cdots\!96\)\( T^{6} + \)\(18\!\cdots\!70\)\( T^{8} - \)\(81\!\cdots\!96\)\( p^{16} T^{10} + 302962005747738748 p^{32} T^{12} - 743676888 p^{48} T^{14} + p^{64} T^{16} )^{2} \)
13 \( ( 1 + 3352372904 T^{2} + 5235706944679709020 T^{4} + \)\(53\!\cdots\!96\)\( T^{6} + \)\(45\!\cdots\!54\)\( T^{8} + \)\(53\!\cdots\!96\)\( p^{16} T^{10} + 5235706944679709020 p^{32} T^{12} + 3352372904 p^{48} T^{14} + p^{64} T^{16} )^{2} \)
17 \( ( 1 - 19363247388 T^{2} + \)\(25\!\cdots\!28\)\( T^{4} - \)\(14\!\cdots\!28\)\( p T^{6} + \)\(20\!\cdots\!70\)\( T^{8} - \)\(14\!\cdots\!28\)\( p^{17} T^{10} + \)\(25\!\cdots\!28\)\( p^{32} T^{12} - 19363247388 p^{48} T^{14} + p^{64} T^{16} )^{2} \)
19 \( ( 1 - 12006 T + 53058888040 T^{2} - 304842858846714 T^{3} + \)\(12\!\cdots\!34\)\( T^{4} - 304842858846714 p^{8} T^{5} + 53058888040 p^{16} T^{6} - 12006 p^{24} T^{7} + p^{32} T^{8} )^{4} \)
23 \( ( 1 - 210185603538 T^{2} + \)\(14\!\cdots\!76\)\( p T^{4} - \)\(34\!\cdots\!46\)\( T^{6} + \)\(58\!\cdots\!30\)\( p^{2} T^{8} - \)\(34\!\cdots\!46\)\( p^{16} T^{10} + \)\(14\!\cdots\!76\)\( p^{33} T^{12} - 210185603538 p^{48} T^{14} + p^{64} T^{16} )^{2} \)
29 \( ( 1 - 1925437662248 T^{2} + \)\(19\!\cdots\!48\)\( T^{4} - \)\(14\!\cdots\!36\)\( T^{6} + \)\(84\!\cdots\!70\)\( T^{8} - \)\(14\!\cdots\!36\)\( p^{16} T^{10} + \)\(19\!\cdots\!48\)\( p^{32} T^{12} - 1925437662248 p^{48} T^{14} + p^{64} T^{16} )^{2} \)
31 \( ( 1 - 117526 T + 1000051193680 T^{2} - 258702506807092234 T^{3} + \)\(14\!\cdots\!94\)\( T^{4} - 258702506807092234 p^{8} T^{5} + 1000051193680 p^{16} T^{6} - 117526 p^{24} T^{7} + p^{32} T^{8} )^{4} \)
37 \( ( 1 + 10932946617352 T^{2} + \)\(63\!\cdots\!28\)\( T^{4} + \)\(31\!\cdots\!84\)\( T^{6} + \)\(13\!\cdots\!70\)\( T^{8} + \)\(31\!\cdots\!84\)\( p^{16} T^{10} + \)\(63\!\cdots\!28\)\( p^{32} T^{12} + 10932946617352 p^{48} T^{14} + p^{64} T^{16} )^{2} \)
41 \( ( 1 - 15942074676228 T^{2} + \)\(21\!\cdots\!08\)\( T^{4} - \)\(15\!\cdots\!16\)\( T^{6} + \)\(13\!\cdots\!70\)\( T^{8} - \)\(15\!\cdots\!16\)\( p^{16} T^{10} + \)\(21\!\cdots\!08\)\( p^{32} T^{12} - 15942074676228 p^{48} T^{14} + p^{64} T^{16} )^{2} \)
43 \( ( 1 + 43842487831054 T^{2} + \)\(10\!\cdots\!60\)\( T^{4} + \)\(17\!\cdots\!66\)\( T^{6} + \)\(23\!\cdots\!94\)\( T^{8} + \)\(17\!\cdots\!66\)\( p^{16} T^{10} + \)\(10\!\cdots\!60\)\( p^{32} T^{12} + 43842487831054 p^{48} T^{14} + p^{64} T^{16} )^{2} \)
47 \( ( 1 - 122512736537778 T^{2} + \)\(72\!\cdots\!28\)\( T^{4} - \)\(27\!\cdots\!86\)\( T^{6} + \)\(74\!\cdots\!70\)\( T^{8} - \)\(27\!\cdots\!86\)\( p^{16} T^{10} + \)\(72\!\cdots\!28\)\( p^{32} T^{12} - 122512736537778 p^{48} T^{14} + p^{64} T^{16} )^{2} \)
53 \( ( 1 - 461042697500828 T^{2} + \)\(95\!\cdots\!28\)\( T^{4} - \)\(11\!\cdots\!36\)\( T^{6} + \)\(88\!\cdots\!70\)\( T^{8} - \)\(11\!\cdots\!36\)\( p^{16} T^{10} + \)\(95\!\cdots\!28\)\( p^{32} T^{12} - 461042697500828 p^{48} T^{14} + p^{64} T^{16} )^{2} \)
59 \( ( 1 - 307604841672408 T^{2} + \)\(88\!\cdots\!88\)\( T^{4} - \)\(17\!\cdots\!16\)\( T^{6} + \)\(29\!\cdots\!70\)\( T^{8} - \)\(17\!\cdots\!16\)\( p^{16} T^{10} + \)\(88\!\cdots\!88\)\( p^{32} T^{12} - 307604841672408 p^{48} T^{14} + p^{64} T^{16} )^{2} \)
61 \( ( 1 - 4712486 T + 520583084559760 T^{2} - \)\(41\!\cdots\!14\)\( T^{3} + \)\(12\!\cdots\!34\)\( T^{4} - \)\(41\!\cdots\!14\)\( p^{8} T^{5} + 520583084559760 p^{16} T^{6} - 4712486 p^{24} T^{7} + p^{32} T^{8} )^{4} \)
67 \( ( 1 + 2815276036900174 T^{2} + \)\(36\!\cdots\!40\)\( T^{4} + \)\(27\!\cdots\!46\)\( T^{6} + \)\(13\!\cdots\!54\)\( T^{8} + \)\(27\!\cdots\!46\)\( p^{16} T^{10} + \)\(36\!\cdots\!40\)\( p^{32} T^{12} + 2815276036900174 p^{48} T^{14} + p^{64} T^{16} )^{2} \)
71 \( ( 1 - 3027552467368328 T^{2} + \)\(47\!\cdots\!28\)\( T^{4} - \)\(49\!\cdots\!36\)\( T^{6} + \)\(37\!\cdots\!70\)\( T^{8} - \)\(49\!\cdots\!36\)\( p^{16} T^{10} + \)\(47\!\cdots\!28\)\( p^{32} T^{12} - 3027552467368328 p^{48} T^{14} + p^{64} T^{16} )^{2} \)
73 \( ( 1 + 1137813863285992 T^{2} + \)\(23\!\cdots\!68\)\( T^{4} + \)\(22\!\cdots\!04\)\( T^{6} + \)\(77\!\cdots\!70\)\( T^{8} + \)\(22\!\cdots\!04\)\( p^{16} T^{10} + \)\(23\!\cdots\!68\)\( p^{32} T^{12} + 1137813863285992 p^{48} T^{14} + p^{64} T^{16} )^{2} \)
79 \( ( 1 - 22694658 T + 3164063420589568 T^{2} - \)\(74\!\cdots\!46\)\( T^{3} + \)\(56\!\cdots\!70\)\( T^{4} - \)\(74\!\cdots\!46\)\( p^{8} T^{5} + 3164063420589568 p^{16} T^{6} - 22694658 p^{24} T^{7} + p^{32} T^{8} )^{4} \)
83 \( ( 1 - 11983081334452338 T^{2} + \)\(71\!\cdots\!28\)\( T^{4} - \)\(27\!\cdots\!26\)\( T^{6} + \)\(73\!\cdots\!70\)\( T^{8} - \)\(27\!\cdots\!26\)\( p^{16} T^{10} + \)\(71\!\cdots\!28\)\( p^{32} T^{12} - 11983081334452338 p^{48} T^{14} + p^{64} T^{16} )^{2} \)
89 \( ( 1 - 15875396463164808 T^{2} + \)\(14\!\cdots\!68\)\( T^{4} - \)\(91\!\cdots\!96\)\( T^{6} + \)\(42\!\cdots\!70\)\( T^{8} - \)\(91\!\cdots\!96\)\( p^{16} T^{10} + \)\(14\!\cdots\!68\)\( p^{32} T^{12} - 15875396463164808 p^{48} T^{14} + p^{64} T^{16} )^{2} \)
97 \( ( 1 + 35638204907329384 T^{2} + \)\(67\!\cdots\!80\)\( T^{4} + \)\(83\!\cdots\!36\)\( T^{6} + \)\(76\!\cdots\!34\)\( T^{8} + \)\(83\!\cdots\!36\)\( p^{16} T^{10} + \)\(67\!\cdots\!80\)\( p^{32} T^{12} + 35638204907329384 p^{48} T^{14} + p^{64} T^{16} )^{2} \)
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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

−2.06669979703095363975280156488, −1.88300568730866401485357005924, −1.86457692166942103550724537827, −1.86281349481692917535223949535, −1.79043983622268165341310894276, −1.53912093856217342318727853558, −1.49703225929375061118708820627, −1.21418133745373966854087963937, −1.17085066500228300414566231229, −1.16880419365591072842263848047, −1.15207599860035991413251206458, −1.13977255189070218217331156523, −1.06673246752448637508787357693, −0.989269967570179794557122510639, −0.965285390194165294142432019198, −0.931351126629499298732798814785, −0.830529694975455498544416170370, −0.73021278735467227023590392100, −0.47999217742439732664105325879, −0.28922435755204425537313193587, −0.25928324358819280471481949548, −0.19528426857499770748336094766, −0.17008238242979917882737652424, −0.11507188354642929694376748798, −0.05648261331035698279185860977, 0.05648261331035698279185860977, 0.11507188354642929694376748798, 0.17008238242979917882737652424, 0.19528426857499770748336094766, 0.25928324358819280471481949548, 0.28922435755204425537313193587, 0.47999217742439732664105325879, 0.73021278735467227023590392100, 0.830529694975455498544416170370, 0.931351126629499298732798814785, 0.965285390194165294142432019198, 0.989269967570179794557122510639, 1.06673246752448637508787357693, 1.13977255189070218217331156523, 1.15207599860035991413251206458, 1.16880419365591072842263848047, 1.17085066500228300414566231229, 1.21418133745373966854087963937, 1.49703225929375061118708820627, 1.53912093856217342318727853558, 1.79043983622268165341310894276, 1.86281349481692917535223949535, 1.86457692166942103550724537827, 1.88300568730866401485357005924, 2.06669979703095363975280156488

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

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