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 − 2·9-s + 4·13-s − 2·17-s + 19-s + 6·23-s − 5·25-s − 5·27-s + 8·29-s − 4·31-s + 2·37-s + 4·39-s − 10·41-s + 8·43-s + 9·47-s − 7·49-s − 2·51-s + 53-s + 57-s + 9·59-s + 10·61-s + 4·67-s + 6·69-s + 12·71-s − 16·73-s − 5·75-s − 13·79-s + ⋯
L(s)  = 1  + 0.577·3-s − 2/3·9-s + 1.10·13-s − 0.485·17-s + 0.229·19-s + 1.25·23-s − 25-s − 0.962·27-s + 1.48·29-s − 0.718·31-s + 0.328·37-s + 0.640·39-s − 1.56·41-s + 1.21·43-s + 1.31·47-s − 49-s − 0.280·51-s + 0.137·53-s + 0.132·57-s + 1.17·59-s + 1.28·61-s + 0.488·67-s + 0.722·69-s + 1.42·71-s − 1.87·73-s − 0.577·75-s − 1.46·79-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 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 13 T + p T^{2} \)
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

−12.98290880000233, −12.47268229264368, −11.77085002263645, −11.47745258190612, −11.08565450058164, −10.64687644386104, −10.01969125795615, −9.635231507957726, −9.056846597143505, −8.588024610978222, −8.437000115757640, −7.937005518650228, −7.169614509237482, −6.932823760373931, −6.315515803440465, −5.680963537542816, −5.492239153074799, −4.768569992677440, −4.083450450637237, −3.790663582790127, −3.078505160753250, −2.743727119227955, −2.139330637269780, −1.428457120188714, −0.8475346113765376, 0, 0.8475346113765376, 1.428457120188714, 2.139330637269780, 2.743727119227955, 3.078505160753250, 3.790663582790127, 4.083450450637237, 4.768569992677440, 5.492239153074799, 5.680963537542816, 6.315515803440465, 6.932823760373931, 7.169614509237482, 7.937005518650228, 8.437000115757640, 8.588024610978222, 9.056846597143505, 9.635231507957726, 10.01969125795615, 10.64687644386104, 11.08565450058164, 11.47745258190612, 11.77085002263645, 12.47268229264368, 12.98290880000233

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