Properties

Label 16-1620e8-1.1-c1e8-0-2
Degree $16$
Conductor $4.744\times 10^{25}$
Sign $1$
Analytic cond. $7.84037\times 10^{8}$
Root an. cond. $3.59663$
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  + 4·7-s + 12·13-s − 10·25-s − 16·31-s − 24·37-s + 12·43-s + 8·49-s + 24·61-s + 4·67-s + 8·73-s + 48·91-s − 36·97-s − 4·103-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + 173-s − 40·175-s + 179-s + ⋯
L(s)  = 1  + 1.51·7-s + 3.32·13-s − 2·25-s − 2.87·31-s − 3.94·37-s + 1.82·43-s + 8/7·49-s + 3.07·61-s + 0.488·67-s + 0.936·73-s + 5.03·91-s − 3.65·97-s − 0.394·103-s − 0.363·121-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 + 5.53·169-s + 0.0760·173-s − 3.02·175-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{32} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(7.84037\times 10^{8}\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1620} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{32} \cdot 5^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.574391747\)
\(L(\frac12)\) \(\approx\) \(2.574391747\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
good7 \( ( 1 - 2 T + 2 T^{2} + 24 T^{3} - 73 T^{4} + 24 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
11 \( ( 1 + 2 T^{2} - 117 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 - 6 T + 18 T^{2} + 48 T^{3} - 313 T^{4} + 48 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
17 \( ( 1 - 2 T^{4} + p^{4} T^{8} )^{2} \)
19 \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \)
23 \( 1 - 238 T^{4} - 223197 T^{8} - 238 p^{4} T^{12} + p^{8} T^{16} \)
29 \( ( 1 - 38 T^{2} + 603 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 7 T + p T^{2} )^{4}( 1 + 11 T + p T^{2} )^{4} \)
37 \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \)
41 \( ( 1 + 2 T^{2} - 1677 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 6 T + 18 T^{2} + 408 T^{3} - 3073 T^{4} + 408 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
47 \( ( 1 - 14 T + 98 T^{2} - 56 T^{3} - 1817 T^{4} - 56 p T^{5} + 98 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )( 1 + 14 T + 98 T^{2} + 56 T^{3} - 1817 T^{4} + 56 p T^{5} + 98 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} ) \)
53 \( ( 1 + 3598 T^{4} + p^{4} T^{8} )^{2} \)
59 \( ( 1 - 38 T^{2} - 2037 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \)
67 \( ( 1 - 2 T + 2 T^{2} + 264 T^{3} - 4753 T^{4} + 264 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
71 \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{4} \)
73 \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \)
79 \( ( 1 + 122 T^{2} + 8643 T^{4} + 122 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
83 \( ( 1 - 22 T + 242 T^{2} - 1672 T^{3} + 11503 T^{4} - 1672 p T^{5} + 242 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} )( 1 + 22 T + 242 T^{2} + 1672 T^{3} + 11503 T^{4} + 1672 p T^{5} + 242 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} ) \)
89 \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{4} \)
97 \( ( 1 + 18 T + 162 T^{2} - 576 T^{3} - 14593 T^{4} - 576 p T^{5} + 162 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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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

−4.03199917230502412674313127492, −3.83596827603476500592920920576, −3.74782906611259078155860522005, −3.67267182813939585496174811113, −3.64064091361791765011527084310, −3.38271765178382891903760379679, −3.23232694926330784702014635706, −3.17155625999351546689529540813, −2.82645155859076089597036765002, −2.80915097450352808230669538043, −2.69549955708055606099580761112, −2.69475329012424432700648197739, −2.33298569583198620094557995712, −2.05057720287685639847181117873, −1.87003445935984427014040164447, −1.86511079933130087026081850102, −1.83337947406733766611022127097, −1.71263847026480758821724003771, −1.57904278080393556445695726078, −1.41878981870276321599380971754, −0.976486658054762368523879259690, −0.957419846163250766956597661213, −0.887878267050731601774804034136, −0.43477835990736845371335197786, −0.13984269534641991482672608736, 0.13984269534641991482672608736, 0.43477835990736845371335197786, 0.887878267050731601774804034136, 0.957419846163250766956597661213, 0.976486658054762368523879259690, 1.41878981870276321599380971754, 1.57904278080393556445695726078, 1.71263847026480758821724003771, 1.83337947406733766611022127097, 1.86511079933130087026081850102, 1.87003445935984427014040164447, 2.05057720287685639847181117873, 2.33298569583198620094557995712, 2.69475329012424432700648197739, 2.69549955708055606099580761112, 2.80915097450352808230669538043, 2.82645155859076089597036765002, 3.17155625999351546689529540813, 3.23232694926330784702014635706, 3.38271765178382891903760379679, 3.64064091361791765011527084310, 3.67267182813939585496174811113, 3.74782906611259078155860522005, 3.83596827603476500592920920576, 4.03199917230502412674313127492

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

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