Properties

Degree 2
Conductor $ 2^{6} \cdot 5077 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

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  =  0
Selberg data  =  $(2,\ 324928,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.978622784$
$L(\frac12)$  $\approx$  $1.978622784$
$L(\frac{3}{2})$   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

−12.59679694312565, −12.14984005532485, −11.78394365689577, −11.15939825144995, −10.78237928922219, −10.28623795164591, −10.08187731541845, −9.428826618326666, −9.031346452914120, −8.384212345087892, −8.163694175210252, −7.494722037337313, −6.780865505404207, −6.620869359893797, −5.999152035235323, −5.450998511755541, −5.172242705610668, −4.853156011879837, −4.009123861519972, −3.421205566909178, −2.767119354481280, −2.382867786536412, −1.817512796218163, −0.9752698874166066, −0.4352046216623160, 0.4352046216623160, 0.9752698874166066, 1.817512796218163, 2.382867786536412, 2.767119354481280, 3.421205566909178, 4.009123861519972, 4.853156011879837, 5.172242705610668, 5.450998511755541, 5.999152035235323, 6.620869359893797, 6.780865505404207, 7.494722037337313, 8.163694175210252, 8.384212345087892, 9.031346452914120, 9.428826618326666, 10.08187731541845, 10.28623795164591, 10.78237928922219, 11.15939825144995, 11.78394365689577, 12.14984005532485, 12.59679694312565

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