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

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s + 10·16-s − 40·43-s − 12·49-s − 20·64-s − 48·79-s + 80·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + 160·172-s + 173-s + 179-s + 181-s + 191-s + 193-s + 48·196-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 2·4-s + 5/2·16-s − 6.09·43-s − 1.71·49-s − 5/2·64-s − 5.40·79-s + 7.27·121-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 + 3.38·169-s + 12.1·172-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 + ⋯

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  =  $(16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $0.7232990304$
$L(\frac12)$  $\approx$  $0.7232990304$
$L(\frac{3}{2})$   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 \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{4} \)
13 \( ( 1 - 22 T^{2} + 259 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
17 \( ( 1 + 29 T^{2} + p^{2} T^{4} )^{4} \)
19 \( ( 1 - 16 T^{2} - 14 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( ( 1 - 45 T^{2} + p^{2} T^{4} )^{4} \)
29 \( ( 1 - 30 T^{2} + 107 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 34 T^{2} + 1411 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 66 T^{2} + p^{2} T^{4} )^{4} \)
41 \( ( 1 + 74 T^{2} + 3931 T^{4} + 74 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 + 10 T + 109 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{4} \)
47 \( ( 1 + 128 T^{2} + 7714 T^{4} + 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 110 T^{2} + 8443 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
59 \( ( 1 + 46 T^{2} + 5691 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 214 T^{2} + 18691 T^{4} - 214 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
67 \( ( 1 + p T^{2} )^{8} \)
71 \( ( 1 - 20 T^{2} - 2618 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 72 T^{2} + 4754 T^{4} - 72 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
79 \( ( 1 + 12 T + 144 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \)
83 \( ( 1 + 62 T^{2} + 9739 T^{4} + 62 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 + 116 T^{2} + 6406 T^{4} + 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 - 48 T^{2} - 9406 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
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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

−3.55898523683382855185636165313, −3.53803757105314416894947795801, −3.47714639900838048032494317869, −3.30313547848045909598391940188, −3.23864499185709903060033516934, −3.06018448128250470943983965555, −2.93787902805643434061578716141, −2.79472200581479824323015195623, −2.70902233278820740219729840214, −2.44049991192942436101943088301, −2.43888306816864550573205535280, −2.31289236571616801977479521157, −2.19840496771697581344328095457, −1.97904439241271793969045367992, −1.62712069405475214258241618563, −1.49848343958977023737306115885, −1.47753366969311752049558230590, −1.44093773270564234418241373701, −1.39831453302952083460859883680, −1.37284302535930975881337069761, −0.890579118909995853677565278460, −0.52007116113276808299272177871, −0.49811295265645591645045166457, −0.29032992962790704553506710299, −0.14658023266366935895210146703, 0.14658023266366935895210146703, 0.29032992962790704553506710299, 0.49811295265645591645045166457, 0.52007116113276808299272177871, 0.890579118909995853677565278460, 1.37284302535930975881337069761, 1.39831453302952083460859883680, 1.44093773270564234418241373701, 1.47753366969311752049558230590, 1.49848343958977023737306115885, 1.62712069405475214258241618563, 1.97904439241271793969045367992, 2.19840496771697581344328095457, 2.31289236571616801977479521157, 2.43888306816864550573205535280, 2.44049991192942436101943088301, 2.70902233278820740219729840214, 2.79472200581479824323015195623, 2.93787902805643434061578716141, 3.06018448128250470943983965555, 3.23864499185709903060033516934, 3.30313547848045909598391940188, 3.47714639900838048032494317869, 3.53803757105314416894947795801, 3.55898523683382855185636165313

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

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