Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 252·3-s + 4.83e3·5-s + 1.67e4·7-s − 1.13e5·9-s − 5.34e5·11-s − 5.77e5·13-s − 1.21e6·15-s − 6.90e6·17-s − 1.06e7·19-s − 4.21e6·21-s − 1.86e7·23-s − 2.54e7·25-s + 7.32e7·27-s + 1.28e8·29-s + 5.28e7·31-s + 1.34e8·33-s + 8.08e7·35-s − 1.82e8·37-s + 1.45e8·39-s + 3.08e8·41-s + 1.71e7·43-s − 5.48e8·45-s − 2.68e9·47-s − 1.69e9·49-s + 1.74e9·51-s − 1.59e9·53-s − 2.58e9·55-s + ⋯
L(s)  = 1  − 0.598·3-s + 0.691·5-s + 0.376·7-s − 0.641·9-s − 1.00·11-s − 0.431·13-s − 0.413·15-s − 1.17·17-s − 0.987·19-s − 0.225·21-s − 0.603·23-s − 0.522·25-s + 0.982·27-s + 1.16·29-s + 0.331·31-s + 0.599·33-s + 0.260·35-s − 0.431·37-s + 0.258·39-s + 0.415·41-s + 0.0177·43-s − 0.443·45-s − 1.70·47-s − 0.858·49-s + 0.706·51-s − 0.524·53-s − 0.691·55-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  =  1
Selberg data  =  $(2,\ 16,\ (\ :11/2),\ -1)$
$L(6)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{13}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p\) is a polynomial of degree 2. If $p = 2$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
good3 \( 1 + 28 p^{2} T + p^{11} T^{2} \)
5 \( 1 - 966 p T + p^{11} T^{2} \)
7 \( 1 - 2392 p T + p^{11} T^{2} \)
11 \( 1 + 534612 T + p^{11} T^{2} \)
13 \( 1 + 577738 T + p^{11} T^{2} \)
17 \( 1 + 6905934 T + p^{11} T^{2} \)
19 \( 1 + 10661420 T + p^{11} T^{2} \)
23 \( 1 + 18643272 T + p^{11} T^{2} \)
29 \( 1 - 128406630 T + p^{11} T^{2} \)
31 \( 1 - 52843168 T + p^{11} T^{2} \)
37 \( 1 + 182213314 T + p^{11} T^{2} \)
41 \( 1 - 308120442 T + p^{11} T^{2} \)
43 \( 1 - 17125708 T + p^{11} T^{2} \)
47 \( 1 + 2687348496 T + p^{11} T^{2} \)
53 \( 1 + 1596055698 T + p^{11} T^{2} \)
59 \( 1 - 5189203740 T + p^{11} T^{2} \)
61 \( 1 - 6956478662 T + p^{11} T^{2} \)
67 \( 1 - 15481826884 T + p^{11} T^{2} \)
71 \( 1 + 9791485272 T + p^{11} T^{2} \)
73 \( 1 - 1463791322 T + p^{11} T^{2} \)
79 \( 1 + 38116845680 T + p^{11} T^{2} \)
83 \( 1 - 29335099668 T + p^{11} T^{2} \)
89 \( 1 + 24992917110 T + p^{11} T^{2} \)
97 \( 1 - 75013568546 T + p^{11} T^{2} \)
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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

−15.86286837037911545729078978023, −14.32530843754175888484820988405, −12.98534730904735475595856359884, −11.42654540973076272090326275745, −10.18498519643619340405857757495, −8.369855434092442029323010829356, −6.33062698752149936323897601978, −4.93479550671668862723293042690, −2.30006357153424662838082619947, 0, 2.30006357153424662838082619947, 4.93479550671668862723293042690, 6.33062698752149936323897601978, 8.369855434092442029323010829356, 10.18498519643619340405857757495, 11.42654540973076272090326275745, 12.98534730904735475595856359884, 14.32530843754175888484820988405, 15.86286837037911545729078978023

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