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  + 640.·3-s + 1.19e4·5-s − 1.74e4·7-s + 2.32e5·9-s − 5.42e5·11-s − 1.08e5·13-s + 7.65e6·15-s + 7.94e6·17-s + 7.86e6·19-s − 1.11e7·21-s − 3.44e7·23-s + 9.40e7·25-s + 3.55e7·27-s − 1.54e8·29-s − 5.14e7·31-s − 3.47e8·33-s − 2.08e8·35-s + 9.22e7·37-s − 6.92e7·39-s + 1.68e8·41-s + 2.39e8·43-s + 2.78e9·45-s + 6.90e7·47-s − 1.67e9·49-s + 5.08e9·51-s − 4.25e9·53-s − 6.48e9·55-s + ⋯
L(s)  = 1  + 1.52·3-s + 1.71·5-s − 0.392·7-s + 1.31·9-s − 1.01·11-s − 0.0808·13-s + 2.60·15-s + 1.35·17-s + 0.729·19-s − 0.597·21-s − 1.11·23-s + 1.92·25-s + 0.476·27-s − 1.39·29-s − 0.323·31-s − 1.54·33-s − 0.671·35-s + 0.218·37-s − 0.122·39-s + 0.226·41-s + 0.248·43-s + 2.24·45-s + 0.0439·47-s − 0.845·49-s + 2.06·51-s − 1.39·53-s − 1.73·55-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  =  $(2,\ 16,\ (\ :11/2),\ 1)$
$L(6)$  $\approx$  $3.46563$
$L(\frac12)$  $\approx$  $3.46563$
$L(\frac{13}{2})$   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 \( 1 - 640.T + 1.77e5T^{2} \)
5 \( 1 - 1.19e4T + 4.88e7T^{2} \)
7 \( 1 + 1.74e4T + 1.97e9T^{2} \)
11 \( 1 + 5.42e5T + 2.85e11T^{2} \)
13 \( 1 + 1.08e5T + 1.79e12T^{2} \)
17 \( 1 - 7.94e6T + 3.42e13T^{2} \)
19 \( 1 - 7.86e6T + 1.16e14T^{2} \)
23 \( 1 + 3.44e7T + 9.52e14T^{2} \)
29 \( 1 + 1.54e8T + 1.22e16T^{2} \)
31 \( 1 + 5.14e7T + 2.54e16T^{2} \)
37 \( 1 - 9.22e7T + 1.77e17T^{2} \)
41 \( 1 - 1.68e8T + 5.50e17T^{2} \)
43 \( 1 - 2.39e8T + 9.29e17T^{2} \)
47 \( 1 - 6.90e7T + 2.47e18T^{2} \)
53 \( 1 + 4.25e9T + 9.26e18T^{2} \)
59 \( 1 - 5.88e9T + 3.01e19T^{2} \)
61 \( 1 - 1.82e9T + 4.35e19T^{2} \)
67 \( 1 + 2.15e10T + 1.22e20T^{2} \)
71 \( 1 + 1.55e10T + 2.31e20T^{2} \)
73 \( 1 - 7.70e9T + 3.13e20T^{2} \)
79 \( 1 - 1.66e9T + 7.47e20T^{2} \)
83 \( 1 - 4.50e10T + 1.28e21T^{2} \)
89 \( 1 + 2.50e10T + 2.77e21T^{2} \)
97 \( 1 - 4.20e10T + 7.15e21T^{2} \)
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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

−16.31116200825258545253202759527, −14.66617974980762707720612933035, −13.78348259345486604673612963891, −12.90621233315029395497971725832, −10.09885135781673194795594083335, −9.356921975928151278472735814093, −7.74152778914749856244192914165, −5.66823903622263311409643752644, −3.08831314804859228294228836739, −1.85827559864606508570080203569, 1.85827559864606508570080203569, 3.08831314804859228294228836739, 5.66823903622263311409643752644, 7.74152778914749856244192914165, 9.356921975928151278472735814093, 10.09885135781673194795594083335, 12.90621233315029395497971725832, 13.78348259345486604673612963891, 14.66617974980762707720612933035, 16.31116200825258545253202759527

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