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Properties

Label 16-3332e8-1.1-c1e8-0-0
Degree $16$
Conductor $1.519\times 10^{28}$
Sign $1$
Analytic cond. $2.51105\times 10^{11}$
Root an. cond. $5.15811$
Motivic weight $1$
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  + 6·9-s − 20·13-s − 12·19-s + 14·25-s − 12·43-s − 24·47-s + 20·53-s + 24·59-s − 4·67-s + 13·81-s − 4·83-s + 12·101-s + 4·103-s − 120·117-s + 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 156·169-s − 72·171-s + 173-s + ⋯
L(s)  = 1  + 2·9-s − 5.54·13-s − 2.75·19-s + 14/5·25-s − 1.82·43-s − 3.50·47-s + 2.74·53-s + 3.12·59-s − 0.488·67-s + 13/9·81-s − 0.439·83-s + 1.19·101-s + 0.394·103-s − 11.0·117-s + 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 12·169-s − 5.50·171-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 7^{16} \cdot 17^{8}\)
Sign: $1$
Analytic conductor: \(2.51105\times 10^{11}\)
Root analytic conductor: \(5.15811\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 7^{16} \cdot 17^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2413517458\)
\(L(\frac12)\) \(\approx\) \(0.2413517458\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
17 \( 1 + 4 T^{2} - 64 T^{3} - 186 T^{4} - 64 p T^{5} + 4 p^{2} T^{6} + p^{4} T^{8} \)
good3 \( 1 - 2 p T^{2} + 23 T^{4} - 44 T^{6} + 31 p T^{8} - 44 p^{2} T^{10} + 23 p^{4} T^{12} - 2 p^{7} T^{14} + p^{8} T^{16} \)
5 \( 1 - 14 T^{2} + 127 T^{4} - 828 T^{6} + 4621 T^{8} - 828 p^{2} T^{10} + 127 p^{4} T^{12} - 14 p^{6} T^{14} + p^{8} T^{16} \)
11 \( 1 - 40 T^{2} + 828 T^{4} - 11800 T^{6} + 138726 T^{8} - 11800 p^{2} T^{10} + 828 p^{4} T^{12} - 40 p^{6} T^{14} + p^{8} T^{16} \)
13 \( ( 1 + 10 T + 72 T^{2} + 366 T^{3} + 1518 T^{4} + 366 p T^{5} + 72 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
19 \( ( 1 + 6 T + 68 T^{2} + 310 T^{3} + 1846 T^{4} + 310 p T^{5} + 68 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 - 60 T^{2} + 1684 T^{4} - 55044 T^{6} + 1650966 T^{8} - 55044 p^{2} T^{10} + 1684 p^{4} T^{12} - 60 p^{6} T^{14} + p^{8} T^{16} \)
29 \( 1 - 168 T^{2} + 13596 T^{4} - 690584 T^{6} + 23965798 T^{8} - 690584 p^{2} T^{10} + 13596 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} \)
31 \( 1 - 214 T^{2} + 20631 T^{4} - 1183996 T^{6} + 44658477 T^{8} - 1183996 p^{2} T^{10} + 20631 p^{4} T^{12} - 214 p^{6} T^{14} + p^{8} T^{16} \)
37 \( 1 - 236 T^{2} + 25540 T^{4} - 1681908 T^{6} + 74692822 T^{8} - 1681908 p^{2} T^{10} + 25540 p^{4} T^{12} - 236 p^{6} T^{14} + p^{8} T^{16} \)
41 \( 1 - 190 T^{2} + 17679 T^{4} - 1073500 T^{6} + 49324909 T^{8} - 1073500 p^{2} T^{10} + 17679 p^{4} T^{12} - 190 p^{6} T^{14} + p^{8} T^{16} \)
43 \( ( 1 + 6 T + 71 T^{2} + 460 T^{3} + 4837 T^{4} + 460 p T^{5} + 71 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
47 \( ( 1 + 12 T + 132 T^{2} + 532 T^{3} + 4326 T^{4} + 532 p T^{5} + 132 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 10 T + 171 T^{2} - 1240 T^{3} + 13065 T^{4} - 1240 p T^{5} + 171 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( ( 1 - 12 T + 112 T^{2} - 812 T^{3} + 5150 T^{4} - 812 p T^{5} + 112 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( 1 - 310 T^{2} + 44871 T^{4} - 4155708 T^{6} + 286150893 T^{8} - 4155708 p^{2} T^{10} + 44871 p^{4} T^{12} - 310 p^{6} T^{14} + p^{8} T^{16} \)
67 \( ( 1 + 2 T + 167 T^{2} - 184 T^{3} + 12613 T^{4} - 184 p T^{5} + 167 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
71 \( 1 - 324 T^{2} + 45108 T^{4} - 3805436 T^{6} + 266696278 T^{8} - 3805436 p^{2} T^{10} + 45108 p^{4} T^{12} - 324 p^{6} T^{14} + p^{8} T^{16} \)
73 \( 1 - 70 T^{2} + 14599 T^{4} - 650716 T^{6} + 91902541 T^{8} - 650716 p^{2} T^{10} + 14599 p^{4} T^{12} - 70 p^{6} T^{14} + p^{8} T^{16} \)
79 \( 1 - 152 T^{2} + 23740 T^{4} - 2698344 T^{6} + 218127622 T^{8} - 2698344 p^{2} T^{10} + 23740 p^{4} T^{12} - 152 p^{6} T^{14} + p^{8} T^{16} \)
83 \( ( 1 + 2 T + 220 T^{2} + 66 T^{3} + 22374 T^{4} + 66 p T^{5} + 220 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
89 \( ( 1 + 116 T^{2} - 1576 T^{3} + 3030 T^{4} - 1576 p T^{5} + 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( 1 - 534 T^{2} + 133399 T^{4} - 20965964 T^{6} + 2355748957 T^{8} - 20965964 p^{2} T^{10} + 133399 p^{4} T^{12} - 534 p^{6} T^{14} + p^{8} T^{16} \)
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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

−3.60849287830368959511386267952, −3.56163266922488827917463520493, −3.21581091017258157656372984246, −3.11962579404281972606828137755, −3.05548079769677507220504438396, −3.02897328914112833376642667710, −2.93056350253628455323185527695, −2.68170589095514221159886427420, −2.48583239761154148714826461522, −2.30235496433900702355281693044, −2.27205195912581843222166120827, −2.26648052045566780643907816648, −2.25864976715427086185058227748, −2.11471390764634898518685891656, −1.98615180281269012984132574129, −1.72353373961591498867974636727, −1.61011208881152902206766143512, −1.46750992795445961672803307247, −1.18793358656584167028619229293, −1.10946044140413153537234159711, −0.951231582620496630672889276670, −0.76864849826291339897850952530, −0.36553688112236375007628074464, −0.34030297114259451989629491535, −0.05999407657252099344839785148, 0.05999407657252099344839785148, 0.34030297114259451989629491535, 0.36553688112236375007628074464, 0.76864849826291339897850952530, 0.951231582620496630672889276670, 1.10946044140413153537234159711, 1.18793358656584167028619229293, 1.46750992795445961672803307247, 1.61011208881152902206766143512, 1.72353373961591498867974636727, 1.98615180281269012984132574129, 2.11471390764634898518685891656, 2.25864976715427086185058227748, 2.26648052045566780643907816648, 2.27205195912581843222166120827, 2.30235496433900702355281693044, 2.48583239761154148714826461522, 2.68170589095514221159886427420, 2.93056350253628455323185527695, 3.02897328914112833376642667710, 3.05548079769677507220504438396, 3.11962579404281972606828137755, 3.21581091017258157656372984246, 3.56163266922488827917463520493, 3.60849287830368959511386267952

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

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