Properties

Label 16-1840e8-1.1-c1e8-0-0
Degree $16$
Conductor $1.314\times 10^{26}$
Sign $1$
Analytic cond. $2.17149\times 10^{9}$
Root an. cond. $3.83307$
Motivic weight $1$
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  − 24·9-s + 20·25-s + 4·29-s − 28·41-s + 18·49-s + 324·81-s + 68·101-s − 88·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 104·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  − 8·9-s + 4·25-s + 0.742·29-s − 4.37·41-s + 18/7·49-s + 36·81-s + 6.76·101-s − 8·121-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 + 8·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 5^{8} \cdot 23^{8}\)
Sign: $1$
Analytic conductor: \(2.17149\times 10^{9}\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1840} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 5^{8} \cdot 23^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.06510761548\)
\(L(\frac12)\) \(\approx\) \(0.06510761548\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( ( 1 - p T^{2} )^{4} \)
23 \( ( 1 + p T^{2} )^{4} \)
good3 \( ( 1 + p T^{2} )^{8} \)
7 \( ( 1 - 9 T^{2} + 32 T^{4} - 9 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
11 \( ( 1 + p T^{2} )^{8} \)
13 \( ( 1 - p T^{2} )^{8} \)
17 \( ( 1 + 11 T^{2} - 168 T^{4} + 11 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( ( 1 + p T^{2} )^{8} \)
29 \( ( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} )^{4} \)
31 \( ( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
37 \( ( 1 + 51 T^{2} + 1232 T^{4} + 51 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
41 \( ( 1 + 7 T + 8 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{4} \)
43 \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{4} \)
47 \( ( 1 + p T^{2} )^{8} \)
53 \( ( 1 - 101 T^{2} + 7392 T^{4} - 101 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
59 \( ( 1 - 11 T + 62 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2}( 1 + 11 T + 62 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \)
61 \( ( 1 - p T^{2} )^{8} \)
67 \( ( 1 + 111 T^{2} + 7832 T^{4} + 111 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
71 \( ( 1 - 13 T + 98 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2}( 1 + 13 T + 98 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \)
73 \( ( 1 - p T^{2} )^{8} \)
79 \( ( 1 + p T^{2} )^{8} \)
83 \( ( 1 - 41 T^{2} - 5208 T^{4} - 41 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 - p T^{2} )^{8} \)
97 \( ( 1 + 174 T^{2} + p^{2} T^{4} )^{4} \)
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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

−3.62539720177268157900757344410, −3.56936192007090404864714354983, −3.55728118899011783371262592912, −3.36359262749177654624770690561, −3.30676245461652783243884568126, −3.15088821386785695523983795958, −3.12659739086198694620414617399, −3.11670386896734681067569997534, −2.97332167050769695731187843206, −2.81794666121454283782123122131, −2.58132555990661226949379539085, −2.46231034709476134061237288727, −2.43061103322267684309924188724, −2.32974750975559208343373195804, −2.31235608551701654622644966725, −1.96457940962829690298066018271, −1.85031029663865860249947389488, −1.78237309686091182520245304873, −1.24964154119273593094300633008, −1.15501308559050647878910335618, −0.842081339616975730092438956600, −0.833040937079540689133397217535, −0.57236667146099631994767377791, −0.30772117322514140432805776825, −0.05081423824520979663742650248, 0.05081423824520979663742650248, 0.30772117322514140432805776825, 0.57236667146099631994767377791, 0.833040937079540689133397217535, 0.842081339616975730092438956600, 1.15501308559050647878910335618, 1.24964154119273593094300633008, 1.78237309686091182520245304873, 1.85031029663865860249947389488, 1.96457940962829690298066018271, 2.31235608551701654622644966725, 2.32974750975559208343373195804, 2.43061103322267684309924188724, 2.46231034709476134061237288727, 2.58132555990661226949379539085, 2.81794666121454283782123122131, 2.97332167050769695731187843206, 3.11670386896734681067569997534, 3.12659739086198694620414617399, 3.15088821386785695523983795958, 3.30676245461652783243884568126, 3.36359262749177654624770690561, 3.55728118899011783371262592912, 3.56936192007090404864714354983, 3.62539720177268157900757344410

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

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