Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·5-s − 2·9-s − 4·11-s + 2·13-s + 2·15-s − 6·17-s + 19-s − 6·23-s − 25-s − 5·27-s + 4·29-s − 8·31-s − 4·33-s + 2·37-s + 2·39-s − 2·43-s − 4·45-s − 7·47-s − 7·49-s − 6·51-s + 9·53-s − 8·55-s + 57-s + 5·59-s − 2·61-s + 4·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s − 2/3·9-s − 1.20·11-s + 0.554·13-s + 0.516·15-s − 1.45·17-s + 0.229·19-s − 1.25·23-s − 1/5·25-s − 0.962·27-s + 0.742·29-s − 1.43·31-s − 0.696·33-s + 0.328·37-s + 0.320·39-s − 0.304·43-s − 0.596·45-s − 1.02·47-s − 49-s − 0.840·51-s + 1.23·53-s − 1.07·55-s + 0.132·57-s + 0.650·59-s − 0.256·61-s + 0.496·65-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  =  2
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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 7 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.14175813127615, −12.91591603502244, −12.17278006198682, −11.57626955346342, −11.27533543463724, −10.75984771568174, −10.24933594181548, −9.903983335429733, −9.411313481076917, −8.817094395253964, −8.619056969780477, −8.018098251027725, −7.718711598528599, −7.000200641602615, −6.527153568110913, −5.934527362777164, −5.641502513887385, −5.192066305577572, −4.486836649019328, −4.006406534020085, −3.350175111287990, −2.790899370737294, −2.346230114360157, −1.916297103383367, −1.333264360439894, 0, 0, 1.333264360439894, 1.916297103383367, 2.346230114360157, 2.790899370737294, 3.350175111287990, 4.006406534020085, 4.486836649019328, 5.192066305577572, 5.641502513887385, 5.934527362777164, 6.527153568110913, 7.000200641602615, 7.718711598528599, 8.018098251027725, 8.619056969780477, 8.817094395253964, 9.411313481076917, 9.903983335429733, 10.24933594181548, 10.75984771568174, 11.27533543463724, 11.57626955346342, 12.17278006198682, 12.91591603502244, 13.14175813127615

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