Properties

Label 16-810e8-1.1-c2e8-0-9
Degree $16$
Conductor $1.853\times 10^{23}$
Sign $1$
Analytic cond. $5.63067\times 10^{10}$
Root an. cond. $4.69796$
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  − 4·4-s + 4·16-s + 96·19-s + 18·25-s + 128·31-s − 128·49-s + 64·61-s + 16·64-s − 384·76-s + 288·79-s − 72·100-s + 640·109-s + 60·121-s − 512·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 676·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 4-s + 1/4·16-s + 5.05·19-s + 0.719·25-s + 4.12·31-s − 2.61·49-s + 1.04·61-s + 1/4·64-s − 5.05·76-s + 3.64·79-s − 0.719·100-s + 5.87·109-s + 0.495·121-s − 4.12·124-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 − 4·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{32} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(5.63067\times 10^{10}\)
Root analytic conductor: \(4.69796\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{32} \cdot 5^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.928397194\)
\(L(\frac12)\) \(\approx\) \(2.928397194\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
3 \( 1 \)
5 \( 1 - 18 T^{2} - 301 T^{4} - 18 p^{4} T^{6} + p^{8} T^{8} \)
good7 \( ( 1 + 64 T^{2} + 1695 T^{4} + 64 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
11 \( ( 1 - 30 T^{2} - 13741 T^{4} - 30 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
13 \( ( 1 - p T + p^{2} T^{2} )^{4}( 1 + p T + p^{2} T^{2} )^{4} \)
17 \( ( 1 + 450 T^{2} + p^{4} T^{4} )^{4} \)
19 \( ( 1 - 12 T + p^{2} T^{2} )^{8} \)
23 \( ( 1 - 480 T^{2} - 49441 T^{4} - 480 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
29 \( ( 1 - p T + p^{2} T^{2} )^{4}( 1 + p T + p^{2} T^{2} )^{4} \)
31 \( ( 1 - 32 T + 63 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
37 \( ( 1 - 2194 T^{2} + p^{4} T^{4} )^{4} \)
41 \( ( 1 + 30 T^{2} - 2824861 T^{4} + 30 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
43 \( ( 1 + 2032 T^{2} + 710223 T^{4} + 2032 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
47 \( ( 1 - 3168 T^{2} + 5156543 T^{4} - 3168 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
53 \( ( 1 + 1010 T^{2} + p^{4} T^{4} )^{4} \)
59 \( ( 1 + 6690 T^{2} + 32638739 T^{4} + 6690 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
61 \( ( 1 - 16 T - 3465 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
67 \( ( 1 + 8944 T^{2} + 59844015 T^{4} + 8944 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
71 \( ( 1 - p T )^{8}( 1 + p T )^{8} \)
73 \( ( 1 + 2942 T^{2} + p^{4} T^{4} )^{4} \)
79 \( ( 1 - 72 T - 1057 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
83 \( ( 1 - 11856 T^{2} + 93106415 T^{4} - 11856 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
89 \( ( 1 - 11490 T^{2} + p^{4} T^{4} )^{4} \)
97 \( ( 1 - 7838 T^{2} - 27095037 T^{4} - 7838 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
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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

−4.29852244240578629994874126854, −3.87628299778469141597971918994, −3.85133355127421471648040228285, −3.84009715178417035810387502491, −3.50868738563641401650157663758, −3.42649152424431394669117587199, −3.36431801859981159797849388747, −3.29911784583145984509211723343, −3.22493994575206901383580772123, −3.05933313280194389489404774109, −2.86834071758835441162461998611, −2.60217272322305084075786901877, −2.39361733205909810586739799038, −2.35091522434072452793597887015, −2.17169935596389801538577890400, −2.16902848157013755140490252538, −1.90333971125888403341631201834, −1.22476300992649258849215577222, −1.21711938909471412987602112658, −1.16274700657797142179421793913, −1.04097387343853413812455076275, −1.01401447120688504147301332463, −0.911650478742015936349948412801, −0.23397663589609460769690268594, −0.20487223769720513872163339257, 0.20487223769720513872163339257, 0.23397663589609460769690268594, 0.911650478742015936349948412801, 1.01401447120688504147301332463, 1.04097387343853413812455076275, 1.16274700657797142179421793913, 1.21711938909471412987602112658, 1.22476300992649258849215577222, 1.90333971125888403341631201834, 2.16902848157013755140490252538, 2.17169935596389801538577890400, 2.35091522434072452793597887015, 2.39361733205909810586739799038, 2.60217272322305084075786901877, 2.86834071758835441162461998611, 3.05933313280194389489404774109, 3.22493994575206901383580772123, 3.29911784583145984509211723343, 3.36431801859981159797849388747, 3.42649152424431394669117587199, 3.50868738563641401650157663758, 3.84009715178417035810387502491, 3.85133355127421471648040228285, 3.87628299778469141597971918994, 4.29852244240578629994874126854

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

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