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Properties

Label 16-3276e8-1.1-c1e8-0-3
Degree $16$
Conductor $1.327\times 10^{28}$
Sign $1$
Analytic cond. $2.19264\times 10^{11}$
Root an. cond. $5.11458$
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  + 2·13-s − 4·17-s + 6·23-s + 21·25-s + 2·29-s − 6·43-s − 4·49-s − 22·53-s + 8·61-s − 26·79-s − 28·101-s − 32·107-s − 50·113-s + 52·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 16·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 0.554·13-s − 0.970·17-s + 1.25·23-s + 21/5·25-s + 0.371·29-s − 0.914·43-s − 4/7·49-s − 3.02·53-s + 1.02·61-s − 2.92·79-s − 2.78·101-s − 3.09·107-s − 4.70·113-s + 4.72·121-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 − 1.23·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{16} \cdot 7^{8} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(2.19264\times 10^{11}\)
Root analytic conductor: \(5.11458\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{16} \cdot 7^{8} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.301236731\)
\(L(\frac12)\) \(\approx\) \(2.301236731\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( ( 1 + T^{2} )^{4} \)
13 \( 1 - 2 T + 20 T^{2} - 94 T^{3} + 278 T^{4} - 94 p T^{5} + 20 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
good5 \( 1 - 21 T^{2} + 249 T^{4} - 2006 T^{6} + 11626 T^{8} - 2006 p^{2} T^{10} + 249 p^{4} T^{12} - 21 p^{6} T^{14} + p^{8} T^{16} \)
11 \( 1 - 52 T^{2} + 1368 T^{4} - 24280 T^{6} + 312502 T^{8} - 24280 p^{2} T^{10} + 1368 p^{4} T^{12} - 52 p^{6} T^{14} + p^{8} T^{16} \)
17 \( ( 1 + 2 T + 16 T^{2} + 24 T^{3} + 10 T^{4} + 24 p T^{5} + 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
19 \( 1 - 81 T^{2} + 3529 T^{4} - 102890 T^{6} + 2243374 T^{8} - 102890 p^{2} T^{10} + 3529 p^{4} T^{12} - 81 p^{6} T^{14} + p^{8} T^{16} \)
23 \( ( 1 - 3 T + 79 T^{2} - 160 T^{3} + 2540 T^{4} - 160 p T^{5} + 79 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - T + 33 T^{2} - 2 T^{3} + 1414 T^{4} - 2 p T^{5} + 33 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( 1 - 125 T^{2} + 8493 T^{4} - 384510 T^{6} + 13439406 T^{8} - 384510 p^{2} T^{10} + 8493 p^{4} T^{12} - 125 p^{6} T^{14} + p^{8} T^{16} \)
37 \( 1 - 84 T^{2} + 2216 T^{4} + 61384 T^{6} - 5827706 T^{8} + 61384 p^{2} T^{10} + 2216 p^{4} T^{12} - 84 p^{6} T^{14} + p^{8} T^{16} \)
41 \( 1 - 172 T^{2} + 16612 T^{4} - 1090964 T^{6} + 51790710 T^{8} - 1090964 p^{2} T^{10} + 16612 p^{4} T^{12} - 172 p^{6} T^{14} + p^{8} T^{16} \)
43 \( ( 1 + 3 T + 87 T^{2} + 20 T^{3} + 4016 T^{4} + 20 p T^{5} + 87 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
47 \( 1 - 137 T^{2} + 13681 T^{4} - 938098 T^{6} + 50458758 T^{8} - 938098 p^{2} T^{10} + 13681 p^{4} T^{12} - 137 p^{6} T^{14} + p^{8} T^{16} \)
53 \( ( 1 + 11 T + 209 T^{2} + 1686 T^{3} + 16482 T^{4} + 1686 p T^{5} + 209 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( 1 - 292 T^{2} + 39492 T^{4} - 3405100 T^{6} + 222712566 T^{8} - 3405100 p^{2} T^{10} + 39492 p^{4} T^{12} - 292 p^{6} T^{14} + p^{8} T^{16} \)
61 \( ( 1 - 4 T + 132 T^{2} - 1068 T^{3} + 8646 T^{4} - 1068 p T^{5} + 132 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 - 168 T^{2} + 18332 T^{4} - 1592024 T^{6} + 121444582 T^{8} - 1592024 p^{2} T^{10} + 18332 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} \)
71 \( 1 - 160 T^{2} + 10920 T^{4} - 867532 T^{6} + 79016406 T^{8} - 867532 p^{2} T^{10} + 10920 p^{4} T^{12} - 160 p^{6} T^{14} + p^{8} T^{16} \)
73 \( 1 - 305 T^{2} + 45549 T^{4} - 4497786 T^{6} + 354140538 T^{8} - 4497786 p^{2} T^{10} + 45549 p^{4} T^{12} - 305 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 + 13 T + 219 T^{2} + 2704 T^{3} + 22084 T^{4} + 2704 p T^{5} + 219 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 117 T^{2} + 15309 T^{4} - 1167406 T^{6} + 97224854 T^{8} - 1167406 p^{2} T^{10} + 15309 p^{4} T^{12} - 117 p^{6} T^{14} + p^{8} T^{16} \)
89 \( 1 - 273 T^{2} + 54405 T^{4} - 7250626 T^{6} + 745491194 T^{8} - 7250626 p^{2} T^{10} + 54405 p^{4} T^{12} - 273 p^{6} T^{14} + p^{8} T^{16} \)
97 \( 1 - 329 T^{2} + 69397 T^{4} - 9804594 T^{6} + 1094792986 T^{8} - 9804594 p^{2} T^{10} + 69397 p^{4} T^{12} - 329 p^{6} T^{14} + p^{8} T^{16} \)
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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

−3.67063405331076510340095833844, −3.39508543152180396243458827032, −3.32089168728891205471997290263, −3.15595059815724933436593019100, −3.04290282253881187147083052461, −2.98453566555373910082216441658, −2.77398016703014255493118419931, −2.72928974954659336799779295943, −2.69180527314960563627560834455, −2.60973049736296790119946064418, −2.47842425404481916264562900416, −2.40342699490560689873850161496, −2.20056182311185154659689413356, −1.91519043399298284939048364015, −1.67770148007758647227028006340, −1.54774990506521917859998408974, −1.53583311093972841324108008246, −1.37637974374407496676731427861, −1.33447250392610322844320176183, −1.28610722456293395051990019899, −0.885989048193185327027851672189, −0.860040924826318634128989553093, −0.52578354477728068428559483920, −0.34562550069372353586130543541, −0.12690236858152335507266055452, 0.12690236858152335507266055452, 0.34562550069372353586130543541, 0.52578354477728068428559483920, 0.860040924826318634128989553093, 0.885989048193185327027851672189, 1.28610722456293395051990019899, 1.33447250392610322844320176183, 1.37637974374407496676731427861, 1.53583311093972841324108008246, 1.54774990506521917859998408974, 1.67770148007758647227028006340, 1.91519043399298284939048364015, 2.20056182311185154659689413356, 2.40342699490560689873850161496, 2.47842425404481916264562900416, 2.60973049736296790119946064418, 2.69180527314960563627560834455, 2.72928974954659336799779295943, 2.77398016703014255493118419931, 2.98453566555373910082216441658, 3.04290282253881187147083052461, 3.15595059815724933436593019100, 3.32089168728891205471997290263, 3.39508543152180396243458827032, 3.67063405331076510340095833844

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

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