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Properties

Label 16-810e8-1.1-c1e8-0-7
Degree $16$
Conductor $1.853\times 10^{23}$
Sign $1$
Analytic cond. $3.06264\times 10^{6}$
Root an. cond. $2.54320$
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  − 12·5-s + 4·7-s − 12·11-s − 12·13-s + 16-s + 36·23-s + 74·25-s − 4·31-s − 48·35-s + 24·41-s + 24·43-s + 36·47-s + 8·49-s + 144·55-s + 24·61-s + 144·65-s − 8·67-s − 16·73-s − 48·77-s − 12·80-s − 12·83-s − 48·91-s − 24·97-s − 12·101-s − 16·103-s + 4·112-s − 24·113-s + ⋯
L(s)  = 1  − 5.36·5-s + 1.51·7-s − 3.61·11-s − 3.32·13-s + 1/4·16-s + 7.50·23-s + 74/5·25-s − 0.718·31-s − 8.11·35-s + 3.74·41-s + 3.65·43-s + 5.25·47-s + 8/7·49-s + 19.4·55-s + 3.07·61-s + 17.8·65-s − 0.977·67-s − 1.87·73-s − 5.47·77-s − 1.34·80-s − 1.31·83-s − 5.03·91-s − 2.43·97-s − 1.19·101-s − 1.57·103-s + 0.377·112-s − 2.25·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{32} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(3.06264\times 10^{6}\)
Root analytic conductor: \(2.54320\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{32} \cdot 5^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.321822140\)
\(L(\frac12)\) \(\approx\) \(1.321822140\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{4} + T^{8} \)
3 \( 1 \)
5 \( ( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
good7 \( ( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} ) \)
11 \( ( 1 + 6 T + 29 T^{2} + 102 T^{3} + 300 T^{4} + 102 p T^{5} + 29 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
13 \( 1 + 12 T + 72 T^{2} + 192 T^{3} - 191 T^{4} - 3672 T^{5} - 11880 T^{6} - 5364 T^{7} + 64080 T^{8} - 5364 p T^{9} - 11880 p^{2} T^{10} - 3672 p^{3} T^{11} - 191 p^{4} T^{12} + 192 p^{5} T^{13} + 72 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 - 802 T^{4} + 296739 T^{8} - 802 p^{4} T^{12} + p^{8} T^{16} \)
19 \( ( 1 - 56 T^{2} + 1410 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( 1 - 36 T + 648 T^{2} - 7776 T^{3} + 70745 T^{4} - 528048 T^{5} + 3400056 T^{6} - 19369332 T^{7} + 98357232 T^{8} - 19369332 p T^{9} + 3400056 p^{2} T^{10} - 528048 p^{3} T^{11} + 70745 p^{4} T^{12} - 7776 p^{5} T^{13} + 648 p^{6} T^{14} - 36 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 - 106 T^{2} + 6769 T^{4} - 295210 T^{6} + 9797332 T^{8} - 295210 p^{2} T^{10} + 6769 p^{4} T^{12} - 106 p^{6} T^{14} + p^{8} T^{16} \)
31 \( ( 1 + 2 T - 53 T^{2} - 10 T^{3} + 2164 T^{4} - 10 p T^{5} - 53 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + p^{2} T^{4} )^{4} \)
41 \( ( 1 - 12 T + 140 T^{2} - 1104 T^{3} + 8751 T^{4} - 1104 p T^{5} + 140 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 - 24 T + 288 T^{2} - 1536 T^{3} - 1895 T^{4} + 85968 T^{5} - 337824 T^{6} - 2751048 T^{7} + 37306944 T^{8} - 2751048 p T^{9} - 337824 p^{2} T^{10} + 85968 p^{3} T^{11} - 1895 p^{4} T^{12} - 1536 p^{5} T^{13} + 288 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \)
47 \( 1 - 36 T + 648 T^{2} - 7776 T^{3} + 71497 T^{4} - 559728 T^{5} + 4053240 T^{6} - 28443924 T^{7} + 196306608 T^{8} - 28443924 p T^{9} + 4053240 p^{2} T^{10} - 559728 p^{3} T^{11} + 71497 p^{4} T^{12} - 7776 p^{5} T^{13} + 648 p^{6} T^{14} - 36 p^{7} T^{15} + p^{8} T^{16} \)
53 \( ( 1 - 3854 T^{4} + p^{4} T^{8} )^{2} \)
59 \( ( 1 - 110 T^{2} + 8619 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 12 T - 8 T^{2} - 360 T^{3} + 10599 T^{4} - 360 p T^{5} - 8 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( ( 1 + 4 T + 8 T^{2} - 504 T^{3} - 5497 T^{4} - 504 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
71 \( ( 1 - 88 T^{2} + 2418 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 + 8 T + 32 T^{2} + 552 T^{3} + 9506 T^{4} + 552 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( 1 + 286 T^{2} + 49081 T^{4} + 5786638 T^{6} + 520832308 T^{8} + 5786638 p^{2} T^{10} + 49081 p^{4} T^{12} + 286 p^{6} T^{14} + p^{8} T^{16} \)
83 \( 1 + 12 T + 72 T^{2} + 288 T^{3} + 10930 T^{4} + 38940 T^{5} - 278208 T^{6} - 269436 T^{7} + 74577699 T^{8} - 269436 p T^{9} - 278208 p^{2} T^{10} + 38940 p^{3} T^{11} + 10930 p^{4} T^{12} + 288 p^{5} T^{13} + 72 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \)
89 \( ( 1 + 40 T^{2} - 5358 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( 1 + 24 T + 288 T^{2} - 624 T^{3} - 53186 T^{4} - 742824 T^{5} - 2315520 T^{6} + 48603528 T^{7} + 893568195 T^{8} + 48603528 p T^{9} - 2315520 p^{2} T^{10} - 742824 p^{3} T^{11} - 53186 p^{4} T^{12} - 624 p^{5} T^{13} + 288 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \)
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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

−4.38406332662316458832842530866, −4.36764073779793897133274954799, −4.30955770495530353951945978342, −4.08364606947413828164963336148, −4.01620376207189498123244245239, −3.94614262438663065432229238476, −3.68428354955759824795175727930, −3.41249586893248342623952728224, −3.37084607146952170137382817161, −2.99683523013772812050494712316, −2.97300194775094667834464751840, −2.75905818124595779049869054261, −2.67795466795192577505938770923, −2.67730359601434678332245413049, −2.58699893337037188300695962643, −2.55351010321204751836851929282, −2.52930594285813096724638692376, −2.00756740948977154329723821611, −1.54358344212619749906001218766, −1.41207077765529495118176223581, −0.974954121314223282874786484061, −0.805782492976644346270709988348, −0.64927785447811514602161469411, −0.59741496628873703188885428697, −0.33343543415276034868301376472, 0.33343543415276034868301376472, 0.59741496628873703188885428697, 0.64927785447811514602161469411, 0.805782492976644346270709988348, 0.974954121314223282874786484061, 1.41207077765529495118176223581, 1.54358344212619749906001218766, 2.00756740948977154329723821611, 2.52930594285813096724638692376, 2.55351010321204751836851929282, 2.58699893337037188300695962643, 2.67730359601434678332245413049, 2.67795466795192577505938770923, 2.75905818124595779049869054261, 2.97300194775094667834464751840, 2.99683523013772812050494712316, 3.37084607146952170137382817161, 3.41249586893248342623952728224, 3.68428354955759824795175727930, 3.94614262438663065432229238476, 4.01620376207189498123244245239, 4.08364606947413828164963336148, 4.30955770495530353951945978342, 4.36764073779793897133274954799, 4.38406332662316458832842530866

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

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