Properties

Degree 2
Conductor $ 2^{4} $
Sign $1$
Motivic weight 11
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 696.·3-s − 4.08e3·5-s − 7.35e4·7-s + 3.07e5·9-s + 3.83e5·11-s + 1.15e6·13-s + 2.84e6·15-s − 6.51e6·17-s + 1.39e7·19-s + 5.12e7·21-s − 1.37e6·23-s − 3.21e7·25-s − 9.07e7·27-s − 7.46e7·29-s − 1.32e7·31-s − 2.66e8·33-s + 3.00e8·35-s − 1.67e7·37-s − 8.06e8·39-s + 1.03e9·41-s − 1.93e8·43-s − 1.25e9·45-s + 1.16e9·47-s + 3.43e9·49-s + 4.53e9·51-s + 4.44e8·53-s − 1.56e9·55-s + ⋯
L(s)  = 1  − 1.65·3-s − 0.584·5-s − 1.65·7-s + 1.73·9-s + 0.717·11-s + 0.865·13-s + 0.966·15-s − 1.11·17-s + 1.29·19-s + 2.73·21-s − 0.0445·23-s − 0.658·25-s − 1.21·27-s − 0.675·29-s − 0.0829·31-s − 1.18·33-s + 0.967·35-s − 0.0396·37-s − 1.43·39-s + 1.39·41-s − 0.201·43-s − 1.01·45-s + 0.737·47-s + 1.73·49-s + 1.84·51-s + 0.145·53-s − 0.419·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(11\)
character  :  $\chi_{16} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 16,\ (\ :11/2),\ 1)$
$L(6)$  $\approx$  $0.568224$
$L(\frac12)$  $\approx$  $0.568224$
$L(\frac{13}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p(T)\) is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + 696.T + 1.77e5T^{2} \)
5 \( 1 + 4.08e3T + 4.88e7T^{2} \)
7 \( 1 + 7.35e4T + 1.97e9T^{2} \)
11 \( 1 - 3.83e5T + 2.85e11T^{2} \)
13 \( 1 - 1.15e6T + 1.79e12T^{2} \)
17 \( 1 + 6.51e6T + 3.42e13T^{2} \)
19 \( 1 - 1.39e7T + 1.16e14T^{2} \)
23 \( 1 + 1.37e6T + 9.52e14T^{2} \)
29 \( 1 + 7.46e7T + 1.22e16T^{2} \)
31 \( 1 + 1.32e7T + 2.54e16T^{2} \)
37 \( 1 + 1.67e7T + 1.77e17T^{2} \)
41 \( 1 - 1.03e9T + 5.50e17T^{2} \)
43 \( 1 + 1.93e8T + 9.29e17T^{2} \)
47 \( 1 - 1.16e9T + 2.47e18T^{2} \)
53 \( 1 - 4.44e8T + 9.26e18T^{2} \)
59 \( 1 - 1.28e8T + 3.01e19T^{2} \)
61 \( 1 - 7.96e9T + 4.35e19T^{2} \)
67 \( 1 - 6.89e9T + 1.22e20T^{2} \)
71 \( 1 - 1.12e10T + 2.31e20T^{2} \)
73 \( 1 - 3.34e9T + 3.13e20T^{2} \)
79 \( 1 + 5.36e10T + 7.47e20T^{2} \)
83 \( 1 - 6.31e10T + 1.28e21T^{2} \)
89 \( 1 - 9.62e10T + 2.77e21T^{2} \)
97 \( 1 + 4.37e10T + 7.15e21T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.27275781335032660734699212824, −15.77208591190569214929515396042, −13.27298414774464976662333642755, −12.06742731671037754752253715619, −11.04009576396169403919370865238, −9.503765499426568174650429211507, −6.91628249184811814931534013839, −5.87315045012548812973032793679, −3.86952645670578587420913653289, −0.61854315435287421392366405863, 0.61854315435287421392366405863, 3.86952645670578587420913653289, 5.87315045012548812973032793679, 6.91628249184811814931534013839, 9.503765499426568174650429211507, 11.04009576396169403919370865238, 12.06742731671037754752253715619, 13.27298414774464976662333642755, 15.77208591190569214929515396042, 16.27275781335032660734699212824

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