Properties

Label 16-12e24-1.1-c2e8-0-4
Degree $16$
Conductor $7.950\times 10^{25}$
Sign $1$
Analytic cond. $2.41562\times 10^{13}$
Root an. cond. $6.86182$
Motivic weight $2$
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  − 104·25-s − 380·49-s − 200·73-s − 568·97-s − 680·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.24e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  − 4.15·25-s − 7.75·49-s − 2.73·73-s − 5.85·97-s − 5.61·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 7.36·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 3^{24}\)
Sign: $1$
Analytic conductor: \(2.41562\times 10^{13}\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 3^{24} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.01915792472\)
\(L(\frac12)\) \(\approx\) \(0.01915792472\)
\(L(2)\) 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 \( 1 \)
good5 \( ( 1 + 26 T^{2} + p^{4} T^{4} )^{4} \)
7 \( ( 1 + 95 T^{2} + p^{4} T^{4} )^{4} \)
11 \( ( 1 + 170 T^{2} + p^{4} T^{4} )^{4} \)
13 \( ( 1 - 311 T^{2} + p^{4} T^{4} )^{4} \)
17 \( ( 1 + 70 T^{2} + p^{4} T^{4} )^{4} \)
19 \( ( 1 - 697 T^{2} + p^{4} T^{4} )^{4} \)
23 \( ( 1 - 458 T^{2} + p^{4} T^{4} )^{4} \)
29 \( ( 1 + 146 T^{2} + p^{4} T^{4} )^{4} \)
31 \( ( 1 + 1730 T^{2} + p^{4} T^{4} )^{4} \)
37 \( ( 1 - 1655 T^{2} + p^{4} T^{4} )^{4} \)
41 \( ( 1 - 3074 T^{2} + p^{4} T^{4} )^{4} \)
43 \( ( 1 - 2542 T^{2} + p^{4} T^{4} )^{4} \)
47 \( ( 1 - 4394 T^{2} + p^{4} T^{4} )^{4} \)
53 \( ( 1 + 5522 T^{2} + p^{4} T^{4} )^{4} \)
59 \( ( 1 + 1130 T^{2} + p^{4} T^{4} )^{4} \)
61 \( ( 1 - 6935 T^{2} + p^{4} T^{4} )^{4} \)
67 \( ( 1 - 8617 T^{2} + p^{4} T^{4} )^{4} \)
71 \( ( 1 - 5378 T^{2} + p^{4} T^{4} )^{4} \)
73 \( ( 1 + 25 T + p^{2} T^{2} )^{8} \)
79 \( ( 1 - 7201 T^{2} + p^{4} T^{4} )^{4} \)
83 \( ( 1 + 3410 T^{2} + p^{4} T^{4} )^{4} \)
89 \( ( 1 - 15770 T^{2} + p^{4} T^{4} )^{4} \)
97 \( ( 1 + 71 T + p^{2} T^{2} )^{8} \)
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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

−3.64357463623479286001161519602, −3.57687976133195778918853748999, −3.51997354301295083176078132139, −3.44534961087600860397628721281, −3.29046534992190201925510319570, −3.08552694168623057054353920629, −2.80248196907302696212327813520, −2.76840751761834773182240751640, −2.75063029059348235117818650543, −2.62938229167158823320327107448, −2.47025765612019520472340109294, −2.41112442751050100054091172338, −2.00316935279502300939463428374, −1.88030231727337441014917715727, −1.79026886765337345102507486722, −1.61894032869533742370819756708, −1.60803613586553424242282463209, −1.41034842124828372133635170184, −1.39079489620171039806435681562, −1.24853302090461583480361544561, −0.878000873276693675359906249040, −0.60290116123172740402373703382, −0.33531751018756129309494605214, −0.099191983489930710560631224315, −0.04243087360422217423076410126, 0.04243087360422217423076410126, 0.099191983489930710560631224315, 0.33531751018756129309494605214, 0.60290116123172740402373703382, 0.878000873276693675359906249040, 1.24853302090461583480361544561, 1.39079489620171039806435681562, 1.41034842124828372133635170184, 1.60803613586553424242282463209, 1.61894032869533742370819756708, 1.79026886765337345102507486722, 1.88030231727337441014917715727, 2.00316935279502300939463428374, 2.41112442751050100054091172338, 2.47025765612019520472340109294, 2.62938229167158823320327107448, 2.75063029059348235117818650543, 2.76840751761834773182240751640, 2.80248196907302696212327813520, 3.08552694168623057054353920629, 3.29046534992190201925510319570, 3.44534961087600860397628721281, 3.51997354301295083176078132139, 3.57687976133195778918853748999, 3.64357463623479286001161519602

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

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