Properties

Label 16-1620e8-1.1-c1e8-0-3
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·7-s − 12·13-s + 8·25-s + 32·31-s − 24·37-s + 32·49-s − 32·67-s + 56·73-s + 96·91-s − 12·97-s + 8·103-s − 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + 173-s − 64·175-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 3.02·7-s − 3.32·13-s + 8/5·25-s + 5.74·31-s − 3.94·37-s + 32/7·49-s − 3.90·67-s + 6.55·73-s + 10.0·91-s − 1.21·97-s + 0.788·103-s − 2.54·121-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 − 4.83·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-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

\(L(1)\) \(\approx\) \(2.390897743\)
\(L(\frac12)\) \(\approx\) \(2.390897743\)
\(L(\frac{3}{2})\) 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 \( 1 - 8 T^{2} + 39 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \)
good7 \( ( 1 + 4 T + 8 T^{2} - 24 T^{3} - 97 T^{4} - 24 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
11 \( ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
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 \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2}( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \)
19 \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \)
23 \( 1 + 734 T^{4} + 258915 T^{8} + 734 p^{4} T^{12} + p^{8} T^{16} \)
29 \( ( 1 + 40 T^{2} + 759 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \)
37 \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \)
41 \( ( 1 + 80 T^{2} + 4719 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2} \)
47 \( 1 + 1918 T^{4} - 1200957 T^{8} + 1918 p^{4} T^{12} + p^{8} T^{16} \)
53 \( ( 1 - 5582 T^{4} + p^{4} T^{8} )^{2} \)
59 \( ( 1 - 110 T^{2} + 8619 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \)
67 \( ( 1 + 16 T + 128 T^{2} - 96 T^{3} - 5257 T^{4} - 96 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
71 \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{4} \)
73 \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{4} \)
79 \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \)
83 \( 1 + 13294 T^{4} + 129272115 T^{8} + 13294 p^{4} T^{12} + p^{8} T^{16} \)
89 \( ( 1 + 176 T^{2} + p^{2} T^{4} )^{4} \)
97 \( ( 1 + 6 T + 18 T^{2} - 1056 T^{3} - 12577 T^{4} - 1056 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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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

−4.11069311061086098812443509234, −3.76458708539468814704926496897, −3.53629304530592602117747161103, −3.45565965013656958940663654478, −3.45535795977494386197080215221, −3.37303052535284331980961969800, −3.27110272424995970412662459316, −2.90139987027001436990363716637, −2.86270285547027462086645026550, −2.78981070836599794742919372380, −2.75923940409879945212821871649, −2.73155364524024758348545088494, −2.43331521916558399397102460263, −2.32255805814521175794380082577, −2.21700420476205526053269984440, −2.18275851933252993978176173930, −1.70185202177236488028992702494, −1.55766333842511828498776983107, −1.51453008937336108994909454266, −1.39070769953331276150063168173, −0.942024902631695657279651541865, −0.63809569458482079840168057041, −0.61630672688956756628791821058, −0.41926565862435733493099776228, −0.27146543327488306504411358977, 0.27146543327488306504411358977, 0.41926565862435733493099776228, 0.61630672688956756628791821058, 0.63809569458482079840168057041, 0.942024902631695657279651541865, 1.39070769953331276150063168173, 1.51453008937336108994909454266, 1.55766333842511828498776983107, 1.70185202177236488028992702494, 2.18275851933252993978176173930, 2.21700420476205526053269984440, 2.32255805814521175794380082577, 2.43331521916558399397102460263, 2.73155364524024758348545088494, 2.75923940409879945212821871649, 2.78981070836599794742919372380, 2.86270285547027462086645026550, 2.90139987027001436990363716637, 3.27110272424995970412662459316, 3.37303052535284331980961969800, 3.45535795977494386197080215221, 3.45565965013656958940663654478, 3.53629304530592602117747161103, 3.76458708539468814704926496897, 4.11069311061086098812443509234

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

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