Properties

Degree 16
Conductor $ 2^{16} \cdot 3^{16} \cdot 11^{8} \cdot 19^{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  − 40·25-s + 56·49-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 14·169-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 + 239-s + 241-s + ⋯
L(s)  = 1  − 8·25-s + 8·49-s − 4·121-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 + 1.07·169-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 + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 11^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 11^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{16} \cdot 3^{16} \cdot 11^{8} \cdot 19^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{7524} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 2^{16} \cdot 3^{16} \cdot 11^{8} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $2.945096751$
$L(\frac12)$  $\approx$  $2.945096751$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;11,\;19\}$, \(F_p\) is a polynomial of degree 16. If $p \in \{2,\;3,\;11,\;19\}$, then $F_p$ is a polynomial of degree at most 15.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 \)
11 \( ( 1 + p T^{2} )^{4} \)
19 \( ( 1 - p T^{2} )^{4} \)
good5 \( ( 1 + p T^{2} )^{8} \)
7 \( ( 1 - p T^{2} )^{8} \)
13 \( ( 1 - 7 T^{2} - 120 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
17 \( ( 1 + 23 T^{2} + 240 T^{4} + 23 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( ( 1 + p T^{2} )^{8} \)
29 \( ( 1 + p T^{2} )^{8} \)
31 \( ( 1 - p T^{2} )^{8} \)
37 \( ( 1 - p T^{2} )^{8} \)
41 \( ( 1 + p T^{2} )^{8} \)
43 \( ( 1 - p T^{2} )^{8} \)
47 \( ( 1 + p T^{2} )^{8} \)
53 \( ( 1 - 103 T^{2} + 7800 T^{4} - 103 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
59 \( ( 1 - 91 T^{2} + 4800 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - p T^{2} )^{8} \)
67 \( ( 1 - p T^{2} )^{8} \)
71 \( ( 1 - 67 T^{2} - 552 T^{4} - 67 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - p T^{2} )^{8} \)
79 \( ( 1 - 139 T^{2} + 13080 T^{4} - 139 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
83 \( ( 1 - 109 T^{2} + 4992 T^{4} - 109 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 - 31 T^{2} - 6960 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
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

−3.37307821788676215199960591400, −2.99525619500179013461861623699, −2.95939563736652677395901390828, −2.78408458280605220181841940106, −2.67688571323049050710653110020, −2.61945232739895613479722508011, −2.48852958374385687561246549992, −2.38119172553117514228671488345, −2.33795129951428205460981606912, −2.14641450627228403128946268609, −2.12940650971654575009921013153, −1.98323766173136007772930153534, −1.93017960363261647183386561467, −1.80100424765510532174729607984, −1.64862461001768516062824284918, −1.54080482326575401819105250747, −1.35613638680366418495357266129, −1.23331505670155620426558959304, −1.02000943263608986425193310825, −0.975413134280614393802005143722, −0.793693308409649463217673520114, −0.53747223767850330091223141646, −0.35544010580295572867287398026, −0.32063942211724448659396539156, −0.14694924778403813915291489496, 0.14694924778403813915291489496, 0.32063942211724448659396539156, 0.35544010580295572867287398026, 0.53747223767850330091223141646, 0.793693308409649463217673520114, 0.975413134280614393802005143722, 1.02000943263608986425193310825, 1.23331505670155620426558959304, 1.35613638680366418495357266129, 1.54080482326575401819105250747, 1.64862461001768516062824284918, 1.80100424765510532174729607984, 1.93017960363261647183386561467, 1.98323766173136007772930153534, 2.12940650971654575009921013153, 2.14641450627228403128946268609, 2.33795129951428205460981606912, 2.38119172553117514228671488345, 2.48852958374385687561246549992, 2.61945232739895613479722508011, 2.67688571323049050710653110020, 2.78408458280605220181841940106, 2.95939563736652677395901390828, 2.99525619500179013461861623699, 3.37307821788676215199960591400

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

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