Properties

Label 16-12e24-1.1-c3e8-0-0
Degree $16$
Conductor $7.950\times 10^{25}$
Sign $1$
Analytic cond. $1.16755\times 10^{16}$
Root an. cond. $10.0972$
Motivic weight $3$
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  − 988·25-s + 2.06e3·49-s + 680·73-s − 3.84e3·97-s + 8.62e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 136·169-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  − 7.90·25-s + 6.02·49-s + 1.09·73-s − 4.02·97-s + 6.47·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.0619·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + 0.000288·229-s + 0.000281·233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 3^{24}\)
Sign: $1$
Analytic conductor: \(1.16755\times 10^{16}\)
Root analytic conductor: \(10.0972\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 3^{24} ,\ ( \ : [3/2]^{8} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.5368346149\)
\(L(\frac12)\) \(\approx\) \(0.5368346149\)
\(L(\frac{5}{2})\) 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 \( ( 1 + 247 T^{2} + p^{6} T^{4} )^{4} \)
7 \( ( 1 - 517 T^{2} + p^{6} T^{4} )^{4} \)
11 \( ( 1 - 2155 T^{2} + p^{6} T^{4} )^{4} \)
13 \( ( 1 + 34 T^{2} + p^{6} T^{4} )^{4} \)
17 \( ( 1 + 3458 T^{2} + p^{6} T^{4} )^{4} \)
19 \( ( 1 + 9290 T^{2} + p^{6} T^{4} )^{4} \)
23 \( ( 1 + 11050 T^{2} + p^{6} T^{4} )^{4} \)
29 \( ( 1 + 5578 T^{2} + p^{6} T^{4} )^{4} \)
31 \( ( 1 - 59413 T^{2} + p^{6} T^{4} )^{4} \)
37 \( ( 1 + 9394 T^{2} + p^{6} T^{4} )^{4} \)
41 \( ( 1 - 446 p T^{2} + p^{6} T^{4} )^{4} \)
43 \( ( 1 + 88166 T^{2} + p^{6} T^{4} )^{4} \)
47 \( ( 1 + 154510 T^{2} + p^{6} T^{4} )^{4} \)
53 \( ( 1 + 285079 T^{2} + p^{6} T^{4} )^{4} \)
59 \( ( 1 - 392506 T^{2} + p^{6} T^{4} )^{4} \)
61 \( ( 1 - 294554 T^{2} + p^{6} T^{4} )^{4} \)
67 \( ( 1 + p^{3} T^{2} )^{8} \)
71 \( ( 1 + 596266 T^{2} + p^{6} T^{4} )^{4} \)
73 \( ( 1 - 85 T + p^{3} T^{2} )^{8} \)
79 \( ( 1 - 273742 T^{2} + p^{6} T^{4} )^{4} \)
83 \( ( 1 + 442013 T^{2} + p^{6} T^{4} )^{4} \)
89 \( ( 1 - 1356802 T^{2} + p^{6} T^{4} )^{4} \)
97 \( ( 1 + 481 T + p^{3} T^{2} )^{8} \)
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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

−3.63951718927737659954455436027, −3.33194824444624480213140316381, −3.30501880586143816774407212537, −3.15501534124972580181745507421, −2.95042601697423530152886012364, −2.90652252529101460933207952662, −2.79539014962157446348843394336, −2.71680962489656428229662058563, −2.38715542644753257783599137692, −2.15913224807574562056287247904, −2.13288162943967496923529929725, −2.07110677527631512525492439589, −2.06309905904355133619892758579, −1.94611204282817469112738662274, −1.72510863958615196056140636645, −1.65728315480646081321187761332, −1.45921153153870503401695301374, −1.23485295467915964572056081016, −0.975697708667852199261069666315, −0.794414056199221766621770249142, −0.77881535465971862402036982340, −0.53086697929975542383088559345, −0.50442400587742024956000183577, −0.16797638864789544153835861607, −0.06602229439426678874178139530, 0.06602229439426678874178139530, 0.16797638864789544153835861607, 0.50442400587742024956000183577, 0.53086697929975542383088559345, 0.77881535465971862402036982340, 0.794414056199221766621770249142, 0.975697708667852199261069666315, 1.23485295467915964572056081016, 1.45921153153870503401695301374, 1.65728315480646081321187761332, 1.72510863958615196056140636645, 1.94611204282817469112738662274, 2.06309905904355133619892758579, 2.07110677527631512525492439589, 2.13288162943967496923529929725, 2.15913224807574562056287247904, 2.38715542644753257783599137692, 2.71680962489656428229662058563, 2.79539014962157446348843394336, 2.90652252529101460933207952662, 2.95042601697423530152886012364, 3.15501534124972580181745507421, 3.30501880586143816774407212537, 3.33194824444624480213140316381, 3.63951718927737659954455436027

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

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