Properties

Degree 16
Conductor $ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} $
Sign $1$
Motivic weight 1
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 + 10·16-s − 12·49-s − 20·64-s − 48·79-s + 80·109-s − 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 24·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 48·196-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  − 2·4-s + 5/2·16-s − 1.71·49-s − 5/2·64-s − 5.40·79-s + 7.66·109-s − 3.63·121-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 + 1.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 24/7·196-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{3150} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $5.380276070$
$L(\frac12)$  $\approx$  $5.380276070$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( ( 1 + T^{2} )^{4} \)
3 \( 1 \)
5 \( 1 \)
7 \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \)
good11 \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \)
13 \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{4} \)
17 \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{4} \)
19 \( ( 1 - 28 T^{2} + p^{2} T^{4} )^{4} \)
23 \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{4} \)
29 \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \)
31 \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \)
37 \( ( 1 - 24 T^{2} + p^{2} T^{4} )^{4} \)
41 \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{4} \)
43 \( ( 1 + 84 T^{2} + p^{2} T^{4} )^{4} \)
47 \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{4} \)
53 \( ( 1 - 14 T + p T^{2} )^{4}( 1 + 14 T + p T^{2} )^{4} \)
59 \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{4} \)
61 \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{4} \)
67 \( ( 1 + 84 T^{2} + p^{2} T^{4} )^{4} \)
71 \( ( 1 - 140 T^{2} + p^{2} T^{4} )^{4} \)
73 \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{4} \)
79 \( ( 1 + 6 T + p T^{2} )^{8} \)
83 \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{4} \)
89 \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{4} \)
97 \( ( 1 - p T^{2} )^{8} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.57420189557019447660713647645, −3.57390549096476523198561226069, −3.46269411584201143085155757548, −3.31464349580203815806681797382, −3.11978808311078708443016738801, −3.11754358940727775660071153124, −2.97491059785818326077798251071, −2.75835059794702199864557875353, −2.56401406705641685592371569511, −2.55541626914829658500895705423, −2.45333222580209387421501968553, −2.35848901994209322197049451166, −2.13393710561386351087995336095, −1.97160056542526262419472543039, −1.69307520812416241718027058058, −1.61269610049448858986037593532, −1.56044560215701069714901800221, −1.46501368366665234254235831081, −1.30234011545016373952410581256, −1.14640946304978255392084492053, −0.880131520107419360271996422045, −0.52446932731773721536870436839, −0.41344192160225552189843759391, −0.40593017970357439573493476341, −0.37633734486404252227810484830, 0.37633734486404252227810484830, 0.40593017970357439573493476341, 0.41344192160225552189843759391, 0.52446932731773721536870436839, 0.880131520107419360271996422045, 1.14640946304978255392084492053, 1.30234011545016373952410581256, 1.46501368366665234254235831081, 1.56044560215701069714901800221, 1.61269610049448858986037593532, 1.69307520812416241718027058058, 1.97160056542526262419472543039, 2.13393710561386351087995336095, 2.35848901994209322197049451166, 2.45333222580209387421501968553, 2.55541626914829658500895705423, 2.56401406705641685592371569511, 2.75835059794702199864557875353, 2.97491059785818326077798251071, 3.11754358940727775660071153124, 3.11978808311078708443016738801, 3.31464349580203815806681797382, 3.46269411584201143085155757548, 3.57390549096476523198561226069, 3.57420189557019447660713647645

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

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