Properties

Label 16-48e8-1.1-c24e8-0-3
Degree $16$
Conductor $2.818\times 10^{13}$
Sign $1$
Analytic cond. $8.87074\times 10^{17}$
Root an. cond. $13.2357$
Motivic weight $24$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.19e5·3-s − 1.09e10·7-s + 3.17e11·9-s + 1.83e13·13-s − 3.13e15·19-s + 5.67e15·21-s + 1.81e17·25-s − 9.72e16·27-s − 8.23e17·31-s + 7.14e18·37-s − 9.51e18·39-s + 1.40e19·43-s − 4.80e20·49-s + 1.63e21·57-s − 2.20e21·61-s − 3.46e21·63-s − 1.55e22·67-s − 4.99e22·73-s − 9.44e22·75-s + 5.99e22·79-s + 2.23e22·81-s − 1.99e23·91-s + 4.28e23·93-s − 2.61e23·97-s + 9.18e23·103-s + 3.72e22·109-s − 3.71e24·111-s + ⋯
L(s)  = 1  − 0.978·3-s − 0.788·7-s + 1.12·9-s + 0.785·13-s − 1.41·19-s + 0.771·21-s + 3.04·25-s − 0.648·27-s − 1.04·31-s + 1.08·37-s − 0.768·39-s + 0.352·43-s − 2.50·49-s + 1.38·57-s − 0.829·61-s − 0.885·63-s − 1.90·67-s − 2.18·73-s − 2.98·75-s + 1.01·79-s + 0.280·81-s − 0.619·91-s + 1.02·93-s − 0.377·97-s + 0.644·103-s + 0.0132·109-s − 1.06·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(25-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+12)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(8.87074\times 10^{17}\)
Root analytic conductor: \(13.2357\)
Motivic weight: \(24\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 3^{8} ,\ ( \ : [12]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{25}{2})\) \(\approx\) \(14.98335390\)
\(L(\frac12)\) \(\approx\) \(14.98335390\)
\(L(13)\) 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 \( 1 + 173320 p T - 64187236 p^{6} T^{2} - 57667685320 p^{13} T^{3} - 152173661066 p^{22} T^{4} - 57667685320 p^{37} T^{5} - 64187236 p^{54} T^{6} + 173320 p^{73} T^{7} + p^{96} T^{8} \)
good5 \( 1 - 7264958854573832 p^{2} T^{2} + \)\(55\!\cdots\!56\)\( p^{5} T^{4} - \)\(34\!\cdots\!32\)\( p^{8} T^{6} + \)\(73\!\cdots\!14\)\( p^{13} T^{8} - \)\(34\!\cdots\!32\)\( p^{56} T^{10} + \)\(55\!\cdots\!56\)\( p^{101} T^{12} - 7264958854573832 p^{146} T^{14} + p^{192} T^{16} \)
7 \( ( 1 + 5455979960 T + 40685035661941198756 p T^{2} + \)\(59\!\cdots\!40\)\( p^{3} T^{3} + \)\(34\!\cdots\!82\)\( p^{6} T^{4} + \)\(59\!\cdots\!40\)\( p^{27} T^{5} + 40685035661941198756 p^{49} T^{6} + 5455979960 p^{72} T^{7} + p^{96} T^{8} )^{2} \)
11 \( 1 - \)\(13\!\cdots\!08\)\( p^{2} T^{2} + \)\(27\!\cdots\!68\)\( p^{3} T^{4} - \)\(23\!\cdots\!16\)\( p^{6} T^{6} + \)\(20\!\cdots\!70\)\( p^{10} T^{8} - \)\(23\!\cdots\!16\)\( p^{54} T^{10} + \)\(27\!\cdots\!68\)\( p^{99} T^{12} - \)\(13\!\cdots\!08\)\( p^{146} T^{14} + p^{192} T^{16} \)
13 \( ( 1 - 9150772373960 T + \)\(20\!\cdots\!72\)\( T^{2} - \)\(99\!\cdots\!40\)\( p T^{3} + \)\(93\!\cdots\!02\)\( p^{2} T^{4} - \)\(99\!\cdots\!40\)\( p^{25} T^{5} + \)\(20\!\cdots\!72\)\( p^{48} T^{6} - 9150772373960 p^{72} T^{7} + p^{96} T^{8} )^{2} \)
17 \( 1 - \)\(66\!\cdots\!28\)\( T^{2} + \)\(18\!\cdots\!24\)\( p T^{4} - \)\(12\!\cdots\!32\)\( p^{3} T^{6} + \)\(13\!\cdots\!10\)\( p^{5} T^{8} - \)\(12\!\cdots\!32\)\( p^{51} T^{10} + \)\(18\!\cdots\!24\)\( p^{97} T^{12} - \)\(66\!\cdots\!28\)\( p^{144} T^{14} + p^{192} T^{16} \)
19 \( ( 1 + 1567577226986456 T + \)\(46\!\cdots\!40\)\( p T^{2} + \)\(95\!\cdots\!64\)\( p^{2} T^{3} + \)\(61\!\cdots\!06\)\( p^{3} T^{4} + \)\(95\!\cdots\!64\)\( p^{26} T^{5} + \)\(46\!\cdots\!40\)\( p^{49} T^{6} + 1567577226986456 p^{72} T^{7} + p^{96} T^{8} )^{2} \)
23 \( 1 - \)\(14\!\cdots\!16\)\( p T^{2} + \)\(39\!\cdots\!24\)\( p^{3} T^{4} - \)\(66\!\cdots\!32\)\( p^{5} T^{6} + \)\(73\!\cdots\!10\)\( p^{7} T^{8} - \)\(66\!\cdots\!32\)\( p^{53} T^{10} + \)\(39\!\cdots\!24\)\( p^{99} T^{12} - \)\(14\!\cdots\!16\)\( p^{145} T^{14} + p^{192} T^{16} \)
29 \( 1 - \)\(55\!\cdots\!08\)\( T^{2} + \)\(11\!\cdots\!68\)\( T^{4} - \)\(93\!\cdots\!96\)\( T^{6} + \)\(52\!\cdots\!70\)\( T^{8} - \)\(93\!\cdots\!96\)\( p^{48} T^{10} + \)\(11\!\cdots\!68\)\( p^{96} T^{12} - \)\(55\!\cdots\!08\)\( p^{144} T^{14} + p^{192} T^{16} \)
31 \( ( 1 + 411750025611247736 T + \)\(20\!\cdots\!20\)\( T^{2} + \)\(80\!\cdots\!84\)\( T^{3} + \)\(17\!\cdots\!94\)\( T^{4} + \)\(80\!\cdots\!84\)\( p^{24} T^{5} + \)\(20\!\cdots\!20\)\( p^{48} T^{6} + 411750025611247736 p^{72} T^{7} + p^{96} T^{8} )^{2} \)
37 \( ( 1 - 3570720321367067720 T + \)\(39\!\cdots\!68\)\( p T^{2} - \)\(49\!\cdots\!80\)\( T^{3} + \)\(90\!\cdots\!06\)\( T^{4} - \)\(49\!\cdots\!80\)\( p^{24} T^{5} + \)\(39\!\cdots\!68\)\( p^{49} T^{6} - 3570720321367067720 p^{72} T^{7} + p^{96} T^{8} )^{2} \)
41 \( 1 - \)\(10\!\cdots\!48\)\( T^{2} + \)\(98\!\cdots\!48\)\( T^{4} - \)\(71\!\cdots\!36\)\( T^{6} + \)\(39\!\cdots\!70\)\( T^{8} - \)\(71\!\cdots\!36\)\( p^{48} T^{10} + \)\(98\!\cdots\!48\)\( p^{96} T^{12} - \)\(10\!\cdots\!48\)\( p^{144} T^{14} + p^{192} T^{16} \)
43 \( ( 1 - 7045275121024018600 T + \)\(26\!\cdots\!12\)\( T^{2} - \)\(94\!\cdots\!00\)\( T^{3} + \)\(40\!\cdots\!38\)\( T^{4} - \)\(94\!\cdots\!00\)\( p^{24} T^{5} + \)\(26\!\cdots\!12\)\( p^{48} T^{6} - 7045275121024018600 p^{72} T^{7} + p^{96} T^{8} )^{2} \)
47 \( 1 - \)\(11\!\cdots\!88\)\( T^{2} + \)\(50\!\cdots\!48\)\( T^{4} - \)\(41\!\cdots\!96\)\( T^{6} + \)\(25\!\cdots\!70\)\( p^{4} T^{8} - \)\(41\!\cdots\!96\)\( p^{48} T^{10} + \)\(50\!\cdots\!48\)\( p^{96} T^{12} - \)\(11\!\cdots\!88\)\( p^{144} T^{14} + p^{192} T^{16} \)
53 \( 1 - \)\(44\!\cdots\!88\)\( T^{2} + \)\(14\!\cdots\!48\)\( T^{4} - \)\(42\!\cdots\!96\)\( T^{6} + \)\(10\!\cdots\!70\)\( T^{8} - \)\(42\!\cdots\!96\)\( p^{48} T^{10} + \)\(14\!\cdots\!48\)\( p^{96} T^{12} - \)\(44\!\cdots\!88\)\( p^{144} T^{14} + p^{192} T^{16} \)
59 \( 1 - \)\(89\!\cdots\!28\)\( T^{2} + \)\(53\!\cdots\!28\)\( T^{4} - \)\(24\!\cdots\!36\)\( T^{6} + \)\(83\!\cdots\!70\)\( T^{8} - \)\(24\!\cdots\!36\)\( p^{48} T^{10} + \)\(53\!\cdots\!28\)\( p^{96} T^{12} - \)\(89\!\cdots\!28\)\( p^{144} T^{14} + p^{192} T^{16} \)
61 \( ( 1 + \)\(11\!\cdots\!64\)\( T + \)\(89\!\cdots\!00\)\( T^{2} + \)\(28\!\cdots\!56\)\( T^{3} + \)\(64\!\cdots\!74\)\( T^{4} + \)\(28\!\cdots\!56\)\( p^{24} T^{5} + \)\(89\!\cdots\!00\)\( p^{48} T^{6} + \)\(11\!\cdots\!64\)\( p^{72} T^{7} + p^{96} T^{8} )^{2} \)
67 \( ( 1 + \)\(77\!\cdots\!80\)\( T + \)\(14\!\cdots\!92\)\( T^{2} + \)\(28\!\cdots\!60\)\( T^{3} + \)\(75\!\cdots\!98\)\( T^{4} + \)\(28\!\cdots\!60\)\( p^{24} T^{5} + \)\(14\!\cdots\!92\)\( p^{48} T^{6} + \)\(77\!\cdots\!80\)\( p^{72} T^{7} + p^{96} T^{8} )^{2} \)
71 \( 1 - \)\(11\!\cdots\!88\)\( T^{2} + \)\(74\!\cdots\!48\)\( T^{4} - \)\(32\!\cdots\!96\)\( T^{6} + \)\(10\!\cdots\!70\)\( T^{8} - \)\(32\!\cdots\!96\)\( p^{48} T^{10} + \)\(74\!\cdots\!48\)\( p^{96} T^{12} - \)\(11\!\cdots\!88\)\( p^{144} T^{14} + p^{192} T^{16} \)
73 \( ( 1 + \)\(24\!\cdots\!60\)\( T + \)\(13\!\cdots\!56\)\( T^{2} + \)\(23\!\cdots\!40\)\( T^{3} + \)\(93\!\cdots\!46\)\( T^{4} + \)\(23\!\cdots\!40\)\( p^{24} T^{5} + \)\(13\!\cdots\!56\)\( p^{48} T^{6} + \)\(24\!\cdots\!60\)\( p^{72} T^{7} + p^{96} T^{8} )^{2} \)
79 \( ( 1 - \)\(29\!\cdots\!12\)\( T + \)\(65\!\cdots\!28\)\( T^{2} - \)\(39\!\cdots\!24\)\( T^{3} + \)\(23\!\cdots\!70\)\( T^{4} - \)\(39\!\cdots\!24\)\( p^{24} T^{5} + \)\(65\!\cdots\!28\)\( p^{48} T^{6} - \)\(29\!\cdots\!12\)\( p^{72} T^{7} + p^{96} T^{8} )^{2} \)
83 \( 1 - \)\(63\!\cdots\!28\)\( T^{2} + \)\(19\!\cdots\!08\)\( T^{4} - \)\(36\!\cdots\!16\)\( T^{6} + \)\(49\!\cdots\!70\)\( T^{8} - \)\(36\!\cdots\!16\)\( p^{48} T^{10} + \)\(19\!\cdots\!08\)\( p^{96} T^{12} - \)\(63\!\cdots\!28\)\( p^{144} T^{14} + p^{192} T^{16} \)
89 \( 1 - \)\(39\!\cdots\!88\)\( T^{2} + \)\(73\!\cdots\!28\)\( T^{4} - \)\(83\!\cdots\!76\)\( T^{6} + \)\(61\!\cdots\!70\)\( T^{8} - \)\(83\!\cdots\!76\)\( p^{48} T^{10} + \)\(73\!\cdots\!28\)\( p^{96} T^{12} - \)\(39\!\cdots\!88\)\( p^{144} T^{14} + p^{192} T^{16} \)
97 \( ( 1 + \)\(13\!\cdots\!80\)\( T + \)\(14\!\cdots\!52\)\( T^{2} + \)\(28\!\cdots\!60\)\( T^{3} + \)\(93\!\cdots\!98\)\( T^{4} + \)\(28\!\cdots\!60\)\( p^{24} T^{5} + \)\(14\!\cdots\!52\)\( p^{48} T^{6} + \)\(13\!\cdots\!80\)\( p^{72} T^{7} + p^{96} T^{8} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.49503461075827833401545056872, −3.49108367820711679998294990782, −3.41591007842847101222120040584, −3.29553029061508267725965178827, −3.22902703549464352818799761449, −3.00033670278953804032575451894, −2.76189001334389377659271908507, −2.72137796800538405746636340495, −2.46779939532781442427785265508, −2.40344229016874364841197972609, −2.17857870863260847701300563035, −2.06026146873614954322923159993, −1.69124606803418345708637356649, −1.67616141932338900084799255507, −1.52737754665953359816330949314, −1.46825949997886577317066294647, −1.38463938132624386934610097681, −1.21414787045539148837372011957, −0.828541692507690423897769459015, −0.76385854597003118528111201914, −0.57198259798077708364615738933, −0.55802196749602096855310954940, −0.41204642900642887372539837654, −0.32287091905691664545611866374, −0.19694475804902186228682879257, 0.19694475804902186228682879257, 0.32287091905691664545611866374, 0.41204642900642887372539837654, 0.55802196749602096855310954940, 0.57198259798077708364615738933, 0.76385854597003118528111201914, 0.828541692507690423897769459015, 1.21414787045539148837372011957, 1.38463938132624386934610097681, 1.46825949997886577317066294647, 1.52737754665953359816330949314, 1.67616141932338900084799255507, 1.69124606803418345708637356649, 2.06026146873614954322923159993, 2.17857870863260847701300563035, 2.40344229016874364841197972609, 2.46779939532781442427785265508, 2.72137796800538405746636340495, 2.76189001334389377659271908507, 3.00033670278953804032575451894, 3.22902703549464352818799761449, 3.29553029061508267725965178827, 3.41591007842847101222120040584, 3.49108367820711679998294990782, 3.49503461075827833401545056872

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

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