Properties

Degree 2
Conductor $ 2^{6} \cdot 503 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 4·5-s − 3·7-s + 6·9-s − 3·11-s − 7·13-s + 12·15-s − 2·17-s − 2·19-s + 9·21-s − 3·23-s + 11·25-s − 9·27-s − 8·29-s − 10·31-s + 9·33-s + 12·35-s − 8·37-s + 21·39-s − 6·41-s − 11·43-s − 24·45-s + 3·47-s + 2·49-s + 6·51-s − 4·53-s + 12·55-s + ⋯
L(s)  = 1  − 1.73·3-s − 1.78·5-s − 1.13·7-s + 2·9-s − 0.904·11-s − 1.94·13-s + 3.09·15-s − 0.485·17-s − 0.458·19-s + 1.96·21-s − 0.625·23-s + 11/5·25-s − 1.73·27-s − 1.48·29-s − 1.79·31-s + 1.56·33-s + 2.02·35-s − 1.31·37-s + 3.36·39-s − 0.937·41-s − 1.67·43-s − 3.57·45-s + 0.437·47-s + 2/7·49-s + 0.840·51-s − 0.549·53-s + 1.61·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 32192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 32192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(32192\)    =    \(2^{6} \cdot 503\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{32192} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(2,\ 32192,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;503\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;503\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
503 \( 1 + T \)
good3 \( 1 + p T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−15.93078954495451, −15.47876655732154, −14.99744735234033, −14.58344699887813, −13.36143316946967, −12.89242262598691, −12.52667901241053, −12.11753207750752, −11.65260375324399, −11.17704365273458, −10.62211124281870, −10.12313855959496, −9.668721858219296, −8.824415938563190, −8.072599920909986, −7.368031051759217, −7.051276581883740, −6.744134464035421, −5.750393784079657, −5.227364066790557, −4.811717316743150, −4.038023685978528, −3.583077188764949, −2.705408730885480, −1.703298968019941, 0, 0, 0, 1.703298968019941, 2.705408730885480, 3.583077188764949, 4.038023685978528, 4.811717316743150, 5.227364066790557, 5.750393784079657, 6.744134464035421, 7.051276581883740, 7.368031051759217, 8.072599920909986, 8.824415938563190, 9.668721858219296, 10.12313855959496, 10.62211124281870, 11.17704365273458, 11.65260375324399, 12.11753207750752, 12.52667901241053, 12.89242262598691, 13.36143316946967, 14.58344699887813, 14.99744735234033, 15.47876655732154, 15.93078954495451

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