Properties

Label 16-3267e8-1.1-c0e8-0-4
Degree $16$
Conductor $1.298\times 10^{28}$
Sign $1$
Analytic cond. $49.9401$
Root an. cond. $1.27688$
Motivic weight $0$
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-s + 5-s + 16-s + 20-s + 8·23-s − 31-s + 2·37-s + 47-s + 49-s − 2·53-s + 59-s + 4·67-s − 2·71-s + 80-s − 16·89-s + 8·92-s − 97-s − 103-s + 113-s + 8·115-s − 124-s + 125-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + ⋯
L(s)  = 1  + 4-s + 5-s + 16-s + 20-s + 8·23-s − 31-s + 2·37-s + 47-s + 49-s − 2·53-s + 59-s + 4·67-s − 2·71-s + 80-s − 16·89-s + 8·92-s − 97-s − 103-s + 113-s + 8·115-s − 124-s + 125-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{24} \cdot 11^{16}\)
Sign: $1$
Analytic conductor: \(49.9401\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3267} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{24} \cdot 11^{16} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.559893368\)
\(L(\frac12)\) \(\approx\) \(5.559893368\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
5 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
7 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
13 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
23 \( ( 1 - T + T^{2} )^{8} \)
29 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
31 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
37 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
41 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
43 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
47 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
53 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
61 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
67 \( ( 1 - T )^{8}( 1 + T + T^{2} )^{4} \)
71 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
79 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
83 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
89 \( ( 1 + T )^{16} \)
97 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
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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

−4.01939688444183918602044045407, −3.45635155482939134579688994336, −3.43077939147794961493488788904, −3.36781643969641202045126232185, −3.12147147766126177954214330555, −3.09851753439624664637011447344, −3.02362605317537372332403189386, −2.97825849042611852837872480355, −2.94747805296358141152182944609, −2.88127019819479572893797242291, −2.62581748094913698932389524942, −2.44720772116325749108722256702, −2.40462589015510938630845445862, −2.26549419501268866170171849637, −2.18730030709731793448207395987, −2.15532165277758020528918703446, −1.67144306035248999820036698691, −1.45490200566881147822610228346, −1.41434960442159240740548448588, −1.38893199992097121535414708850, −1.26888333004811860866723755382, −1.26441198886815818114745929824, −0.994947766909477974499249911054, −0.794313357271090733583329391944, −0.49757716454336614095278597182, 0.49757716454336614095278597182, 0.794313357271090733583329391944, 0.994947766909477974499249911054, 1.26441198886815818114745929824, 1.26888333004811860866723755382, 1.38893199992097121535414708850, 1.41434960442159240740548448588, 1.45490200566881147822610228346, 1.67144306035248999820036698691, 2.15532165277758020528918703446, 2.18730030709731793448207395987, 2.26549419501268866170171849637, 2.40462589015510938630845445862, 2.44720772116325749108722256702, 2.62581748094913698932389524942, 2.88127019819479572893797242291, 2.94747805296358141152182944609, 2.97825849042611852837872480355, 3.02362605317537372332403189386, 3.09851753439624664637011447344, 3.12147147766126177954214330555, 3.36781643969641202045126232185, 3.43077939147794961493488788904, 3.45635155482939134579688994336, 4.01939688444183918602044045407

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

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