Properties

Label 16-3360e8-1.1-c0e8-0-3
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·25-s − 2·81-s − 12·89-s + 12·101-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-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  + 2·25-s − 2·81-s − 12·89-s + 12·101-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-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 + ⋯

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: induced by $\chi_{3360} (1, \cdot )$
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

\(L(\frac{1}{2})\) \(\approx\) \(2.102743027\)
\(L(\frac12)\) \(\approx\) \(2.102743027\)
\(L(1)\) 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 \( ( 1 + T^{4} )^{2} \)
5 \( ( 1 - T^{2} + T^{4} )^{2} \)
7 \( 1 - T^{4} + T^{8} \)
good11 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
13 \( ( 1 - T )^{8}( 1 + T )^{8} \)
17 \( ( 1 - T^{2} + T^{4} )^{4} \)
19 \( ( 1 - T^{2} + T^{4} )^{4} \)
23 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
29 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
31 \( ( 1 - T^{2} + T^{4} )^{4} \)
37 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
41 \( ( 1 - T^{2} + T^{4} )^{4} \)
43 \( ( 1 - T^{4} + T^{8} )^{2} \)
47 \( ( 1 - T^{4} + T^{8} )^{2} \)
53 \( ( 1 - T^{2} + T^{4} )^{4} \)
59 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
61 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
67 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
71 \( ( 1 - T )^{8}( 1 + T )^{8} \)
73 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
79 \( ( 1 - T^{2} + T^{4} )^{4} \)
83 \( ( 1 - T^{4} + T^{8} )^{2} \)
89 \( ( 1 + T )^{8}( 1 + T + T^{2} )^{4} \)
97 \( ( 1 - T )^{8}( 1 + T )^{8} \)
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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

−3.77116944340870452643935491305, −3.73616002315054518555519974145, −3.50958670979317578234957647762, −3.47535536259215937973744419663, −3.26094377914878128619017677329, −3.04911743384282214052360479200, −3.03193268549033210467863904678, −3.02664281931911211285422845476, −2.84223756950420830526606167491, −2.75654204861734006317750615036, −2.67187324296023750416213216565, −2.57072517571473936731330734536, −2.44449803557646173338612020518, −2.20379385791908296630401422755, −2.05588211203471848073006824844, −1.85402857033447995527481432060, −1.71639902272188519168533392137, −1.64990733825982099400825637658, −1.56593148908373671913782018166, −1.56159748332681379007208729406, −1.18001691211894767998804880168, −1.00817362582173333841271347532, −0.827950920659224598987851089116, −0.66090180338384530802245674140, −0.42669062067821938387986788612, 0.42669062067821938387986788612, 0.66090180338384530802245674140, 0.827950920659224598987851089116, 1.00817362582173333841271347532, 1.18001691211894767998804880168, 1.56159748332681379007208729406, 1.56593148908373671913782018166, 1.64990733825982099400825637658, 1.71639902272188519168533392137, 1.85402857033447995527481432060, 2.05588211203471848073006824844, 2.20379385791908296630401422755, 2.44449803557646173338612020518, 2.57072517571473936731330734536, 2.67187324296023750416213216565, 2.75654204861734006317750615036, 2.84223756950420830526606167491, 3.02664281931911211285422845476, 3.03193268549033210467863904678, 3.04911743384282214052360479200, 3.26094377914878128619017677329, 3.47535536259215937973744419663, 3.50958670979317578234957647762, 3.73616002315054518555519974145, 3.77116944340870452643935491305

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

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