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  + 3·3-s + 4·5-s − 4·7-s + 6·9-s + 6·11-s + 4·13-s + 12·15-s − 4·17-s + 7·19-s − 12·21-s − 6·23-s + 11·25-s + 9·27-s + 6·29-s − 2·31-s + 18·33-s − 16·35-s + 12·39-s + 8·43-s + 24·45-s − 9·47-s + 9·49-s − 12·51-s + 9·53-s + 24·55-s + 21·57-s + 11·59-s + ⋯
L(s)  = 1  + 1.73·3-s + 1.78·5-s − 1.51·7-s + 2·9-s + 1.80·11-s + 1.10·13-s + 3.09·15-s − 0.970·17-s + 1.60·19-s − 2.61·21-s − 1.25·23-s + 11/5·25-s + 1.73·27-s + 1.11·29-s − 0.359·31-s + 3.13·33-s − 2.70·35-s + 1.92·39-s + 1.21·43-s + 3.57·45-s − 1.31·47-s + 9/7·49-s − 1.68·51-s + 1.23·53-s + 3.23·55-s + 2.78·57-s + 1.43·59-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  =  $(2,\ 324928,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $11.74274690$
$L(\frac12)$  $\approx$  $11.74274690$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
5077 \( 1 - T \)
good3 \( 1 - p T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
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

−12.96503093420736, −12.26172487999220, −11.88829681700848, −11.20942732304584, −10.46742788115159, −10.04586848163279, −9.709172828291640, −9.339880632869854, −9.153023921144892, −8.608787014050755, −8.346067918064679, −7.418934069241798, −6.937609287588536, −6.555551397434359, −6.196625546337939, −5.802710254527197, −5.076603462729062, −4.196513641422559, −3.843655152290611, −3.409413252173944, −2.884120901774073, −2.382553389089872, −1.880141747678526, −1.285087031077933, −0.8310645816144157, 0.8310645816144157, 1.285087031077933, 1.880141747678526, 2.382553389089872, 2.884120901774073, 3.409413252173944, 3.843655152290611, 4.196513641422559, 5.076603462729062, 5.802710254527197, 6.196625546337939, 6.555551397434359, 6.937609287588536, 7.418934069241798, 8.346067918064679, 8.608787014050755, 9.153023921144892, 9.339880632869854, 9.709172828291640, 10.04586848163279, 10.46742788115159, 11.20942732304584, 11.88829681700848, 12.26172487999220, 12.96503093420736

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