Properties

Degree 2
Conductor $ 2^{6} \cdot 5077 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·5-s − 2·9-s + 2·11-s − 2·13-s + 2·15-s + 4·17-s + 5·19-s − 6·23-s − 25-s − 5·27-s + 6·29-s + 2·31-s + 2·33-s − 4·37-s − 2·39-s + 2·41-s + 10·43-s − 4·45-s − 11·47-s − 7·49-s + 4·51-s + 5·53-s + 4·55-s + 5·57-s − 11·59-s + 2·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s − 2/3·9-s + 0.603·11-s − 0.554·13-s + 0.516·15-s + 0.970·17-s + 1.14·19-s − 1.25·23-s − 1/5·25-s − 0.962·27-s + 1.11·29-s + 0.359·31-s + 0.348·33-s − 0.657·37-s − 0.320·39-s + 0.312·41-s + 1.52·43-s − 0.596·45-s − 1.60·47-s − 49-s + 0.560·51-s + 0.686·53-s + 0.539·55-s + 0.662·57-s − 1.43·59-s + 0.256·61-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 324928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 324928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(324928\)    =    \(2^{6} \cdot 5077\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{324928} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 324928,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5077\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5077\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5077 \( 1 + T \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 11 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.79580231156361, −12.40760372155893, −11.80713631238723, −11.69759208792083, −11.07901916569100, −10.31450351661276, −10.01377953119153, −9.646405404375203, −9.311746484041524, −8.699985392405196, −8.304808760352327, −7.767209741648385, −7.416034435090561, −6.804738719464547, −6.113089308627176, −5.860193976874653, −5.478540814414174, −4.751605425331361, −4.334395789451939, −3.536271768668002, −3.132535001782616, −2.712736548475900, −2.014531272379312, −1.576308120788072, −0.8957325301989148, 0, 0.8957325301989148, 1.576308120788072, 2.014531272379312, 2.712736548475900, 3.132535001782616, 3.536271768668002, 4.334395789451939, 4.751605425331361, 5.478540814414174, 5.860193976874653, 6.113089308627176, 6.804738719464547, 7.416034435090561, 7.767209741648385, 8.304808760352327, 8.699985392405196, 9.311746484041524, 9.646405404375203, 10.01377953119153, 10.31450351661276, 11.07901916569100, 11.69759208792083, 11.80713631238723, 12.40760372155893, 12.79580231156361

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