Properties

Label 16-42e16-1.1-c1e8-0-3
Degree $16$
Conductor $9.375\times 10^{25}$
Sign $1$
Analytic cond. $1.54954\times 10^{9}$
Root an. cond. $3.75308$
Motivic weight $1$
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  + 2·2-s + 3·4-s + 2·8-s + 3·16-s + 12·19-s + 18·25-s + 16·29-s − 12·31-s + 6·32-s − 12·37-s + 24·38-s + 8·47-s + 36·50-s − 8·53-s + 32·58-s − 28·59-s − 24·62-s + 11·64-s − 24·74-s + 36·76-s − 4·83-s + 16·94-s + 54·100-s − 20·103-s − 16·106-s − 12·109-s − 32·113-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.707·8-s + 3/4·16-s + 2.75·19-s + 18/5·25-s + 2.97·29-s − 2.15·31-s + 1.06·32-s − 1.97·37-s + 3.89·38-s + 1.16·47-s + 5.09·50-s − 1.09·53-s + 4.20·58-s − 3.64·59-s − 3.04·62-s + 11/8·64-s − 2.78·74-s + 4.12·76-s − 0.439·83-s + 1.65·94-s + 27/5·100-s − 1.97·103-s − 1.55·106-s − 1.14·109-s − 3.01·113-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{16} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(1.54954\times 10^{9}\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1764} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(11.00508415\)
\(L(\frac12)\) \(\approx\) \(11.00508415\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T + T^{2} + p T^{3} - 3 p T^{4} + p^{2} T^{5} + p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 18 T^{2} + 153 T^{4} - 906 T^{6} + 4676 T^{8} - 906 p^{2} T^{10} + 153 p^{4} T^{12} - 18 p^{6} T^{14} + p^{8} T^{16} \)
11 \( ( 1 - 10 T + 25 T^{2} + 94 T^{3} - 684 T^{4} + 94 p T^{5} + 25 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )( 1 + 10 T + 25 T^{2} - 94 T^{3} - 684 T^{4} - 94 p T^{5} + 25 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} ) \)
13 \( 1 - 66 T^{2} + 2241 T^{4} - 49362 T^{6} + 759092 T^{8} - 49362 p^{2} T^{10} + 2241 p^{4} T^{12} - 66 p^{6} T^{14} + p^{8} T^{16} \)
17 \( 1 - 80 T^{2} + 3228 T^{4} - 87472 T^{6} + 1728326 T^{8} - 87472 p^{2} T^{10} + 3228 p^{4} T^{12} - 80 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 - 6 T + 69 T^{2} - 282 T^{3} + 1904 T^{4} - 282 p T^{5} + 69 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 - 104 T^{2} + 5628 T^{4} - 203992 T^{6} + 5416646 T^{8} - 203992 p^{2} T^{10} + 5628 p^{4} T^{12} - 104 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 - 8 T + 71 T^{2} - 344 T^{3} + 1924 T^{4} - 344 p T^{5} + 71 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 6 T + 40 T^{2} + 288 T^{3} + 2601 T^{4} + 288 p T^{5} + 40 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 6 T + 105 T^{2} + 306 T^{3} + 4436 T^{4} + 306 p T^{5} + 105 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( 1 - 120 T^{2} + 9948 T^{4} - 607176 T^{6} + 27583238 T^{8} - 607176 p^{2} T^{10} + 9948 p^{4} T^{12} - 120 p^{6} T^{14} + p^{8} T^{16} \)
43 \( 1 - 210 T^{2} + 23793 T^{4} - 1729746 T^{6} + 88400276 T^{8} - 1729746 p^{2} T^{10} + 23793 p^{4} T^{12} - 210 p^{6} T^{14} + p^{8} T^{16} \)
47 \( ( 1 - 4 T + 160 T^{2} - 548 T^{3} + 10686 T^{4} - 548 p T^{5} + 160 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
53 \( ( 1 + 4 T + 151 T^{2} + 752 T^{3} + 10380 T^{4} + 752 p T^{5} + 151 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( ( 1 + 14 T + 209 T^{2} + 2018 T^{3} + 18892 T^{4} + 2018 p T^{5} + 209 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( 1 - 216 T^{2} + 27420 T^{4} - 2531496 T^{6} + 176195558 T^{8} - 2531496 p^{2} T^{10} + 27420 p^{4} T^{12} - 216 p^{6} T^{14} + p^{8} T^{16} \)
67 \( 1 - 282 T^{2} + 45393 T^{4} - 4904034 T^{6} + 383059316 T^{8} - 4904034 p^{2} T^{10} + 45393 p^{4} T^{12} - 282 p^{6} T^{14} + p^{8} T^{16} \)
71 \( 1 - 288 T^{2} + 41724 T^{4} - 4336608 T^{6} + 350671046 T^{8} - 4336608 p^{2} T^{10} + 41724 p^{4} T^{12} - 288 p^{6} T^{14} + p^{8} T^{16} \)
73 \( 1 - 370 T^{2} + 70161 T^{4} - 8626802 T^{6} + 743765828 T^{8} - 8626802 p^{2} T^{10} + 70161 p^{4} T^{12} - 370 p^{6} T^{14} + p^{8} T^{16} \)
79 \( 1 - 228 T^{2} + 23202 T^{4} - 2215104 T^{6} + 207545771 T^{8} - 2215104 p^{2} T^{10} + 23202 p^{4} T^{12} - 228 p^{6} T^{14} + p^{8} T^{16} \)
83 \( ( 1 + 2 T + 229 T^{2} + 802 T^{3} + 24432 T^{4} + 802 p T^{5} + 229 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
89 \( 1 - 528 T^{2} + 131868 T^{4} - 20568048 T^{6} + 2191026566 T^{8} - 20568048 p^{2} T^{10} + 131868 p^{4} T^{12} - 528 p^{6} T^{14} + p^{8} T^{16} \)
97 \( 1 - 594 T^{2} + 165777 T^{4} - 28537554 T^{6} + 3324136868 T^{8} - 28537554 p^{2} T^{10} + 165777 p^{4} T^{12} - 594 p^{6} T^{14} + p^{8} T^{16} \)
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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

−3.92351726838100883168858161247, −3.87614440003906507050441646544, −3.62296882261664716608513027727, −3.33553128952847161929662951067, −3.32774851587226308112256179170, −3.30876120807947010944703538173, −3.21167633678758400375125305966, −3.12670426843768229279322485222, −2.94531777424141622825843341362, −2.82145891216457744108759035923, −2.75376166342849971302876794000, −2.65127594981485515581615142920, −2.43269225113038701698096456837, −2.38358945119011199434539036595, −2.23241310953148951405177037529, −1.79957215052806624055350374809, −1.68575329317485246365903913710, −1.60129055609798125631774415925, −1.47937107624136169135333741134, −1.32402907241858470187908466907, −1.17803378896638368248984709221, −0.889602165898357647300830584059, −0.825218718624443141646093779252, −0.54722814817202318657917996713, −0.16881553361392887664000249791, 0.16881553361392887664000249791, 0.54722814817202318657917996713, 0.825218718624443141646093779252, 0.889602165898357647300830584059, 1.17803378896638368248984709221, 1.32402907241858470187908466907, 1.47937107624136169135333741134, 1.60129055609798125631774415925, 1.68575329317485246365903913710, 1.79957215052806624055350374809, 2.23241310953148951405177037529, 2.38358945119011199434539036595, 2.43269225113038701698096456837, 2.65127594981485515581615142920, 2.75376166342849971302876794000, 2.82145891216457744108759035923, 2.94531777424141622825843341362, 3.12670426843768229279322485222, 3.21167633678758400375125305966, 3.30876120807947010944703538173, 3.32774851587226308112256179170, 3.33553128952847161929662951067, 3.62296882261664716608513027727, 3.87614440003906507050441646544, 3.92351726838100883168858161247

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

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