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Properties

Label 16-3360e8-1.1-c0e8-0-0
Degree $16$
Conductor $1.624\times 10^{28}$
Sign $1$
Analytic cond. $62.5131$
Root an. cond. $1.29493$
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  + 2·49-s − 4·73-s − 12·79-s + 81-s − 4·97-s + 4·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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  + 2·49-s − 4·73-s − 12·79-s + 81-s − 4·97-s + 4·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{40} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(62.5131\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{40} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.219953177\)
\(L(\frac12)\) \(\approx\) \(1.219953177\)
\(L(1)\) 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 \( 1 - T^{4} + T^{8} \)
5 \( ( 1 + T^{4} )^{2} \)
7 \( ( 1 - T^{2} + T^{4} )^{2} \)
good11 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
13 \( ( 1 + T^{4} )^{4} \)
17 \( ( 1 - T^{4} + T^{8} )^{2} \)
19 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
23 \( ( 1 - T^{4} + T^{8} )^{2} \)
29 \( ( 1 - T^{4} + T^{8} )^{2} \)
31 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
37 \( ( 1 - T^{4} + T^{8} )^{2} \)
41 \( ( 1 + T^{2} )^{8} \)
43 \( ( 1 + T^{4} )^{4} \)
47 \( ( 1 - T^{4} + T^{8} )^{2} \)
53 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
59 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
61 \( ( 1 - T^{2} + T^{4} )^{4} \)
67 \( ( 1 - T^{4} + T^{8} )^{2} \)
71 \( ( 1 - T )^{8}( 1 + T )^{8} \)
73 \( ( 1 + T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
79 \( ( 1 + T )^{8}( 1 + T + T^{2} )^{4} \)
83 \( ( 1 - T^{4} + T^{8} )^{2} \)
89 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
97 \( ( 1 + T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
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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

−3.84515637862489627195311318068, −3.69326227544887647751906656213, −3.61226466163000569988799377879, −3.48501601351933442731974813860, −3.08810853831929555066588952458, −2.98485407401563782897383173469, −2.97466540525549432201677106439, −2.91834510710367477484202101888, −2.91537750021228595226378109068, −2.90329777599707601540219528673, −2.77491066336319699064853449916, −2.68330665081589269561858035395, −2.20798261251585700494844365503, −2.18413776294159165900160428085, −2.08048238113798177668263172539, −1.85552736412119374360199749842, −1.76124191434841000765943120228, −1.71607228453058383540700425256, −1.67449521619124671966554640721, −1.31182727849646702546286347396, −1.13623589518279083939286337176, −1.12441256616508916857645788890, −1.01600323489916213778506910091, −0.49696285530254795849518448935, −0.35356353233888616892835095014, 0.35356353233888616892835095014, 0.49696285530254795849518448935, 1.01600323489916213778506910091, 1.12441256616508916857645788890, 1.13623589518279083939286337176, 1.31182727849646702546286347396, 1.67449521619124671966554640721, 1.71607228453058383540700425256, 1.76124191434841000765943120228, 1.85552736412119374360199749842, 2.08048238113798177668263172539, 2.18413776294159165900160428085, 2.20798261251585700494844365503, 2.68330665081589269561858035395, 2.77491066336319699064853449916, 2.90329777599707601540219528673, 2.91537750021228595226378109068, 2.91834510710367477484202101888, 2.97466540525549432201677106439, 2.98485407401563782897383173469, 3.08810853831929555066588952458, 3.48501601351933442731974813860, 3.61226466163000569988799377879, 3.69326227544887647751906656213, 3.84515637862489627195311318068

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

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