Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 11^{2} $
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 + 2·7-s + 9-s − 2·13-s + 2·15-s − 4·17-s − 6·19-s − 2·21-s − 25-s − 27-s − 8·29-s − 8·31-s − 4·35-s − 10·37-s + 2·39-s − 8·41-s − 2·43-s − 2·45-s − 8·47-s − 3·49-s + 4·51-s + 2·53-s + 6·57-s − 12·59-s + 10·61-s + 2·63-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 0.755·7-s + 1/3·9-s − 0.554·13-s + 0.516·15-s − 0.970·17-s − 1.37·19-s − 0.436·21-s − 1/5·25-s − 0.192·27-s − 1.48·29-s − 1.43·31-s − 0.676·35-s − 1.64·37-s + 0.320·39-s − 1.24·41-s − 0.304·43-s − 0.298·45-s − 1.16·47-s − 3/7·49-s + 0.560·51-s + 0.274·53-s + 0.794·57-s − 1.56·59-s + 1.28·61-s + 0.251·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(23232\)    =    \(2^{6} \cdot 3 \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{23232} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(2,\ 23232,\ (\ :1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
11 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−16.07151343376059, −15.31925932656467, −14.93715413369038, −14.73202675395121, −13.75243125781431, −13.27445027919837, −12.60203664300314, −12.20142037339500, −11.57023192490572, −11.09063938409442, −10.81678409657385, −10.10109620827608, −9.342000473839292, −8.721712191905006, −8.191645328076428, −7.609886518008811, −6.987916175323969, −6.560463682239176, −5.639636753851219, −5.086763598563966, −4.494675021618772, −3.924558228805985, −3.274347855143445, −1.986374781128185, −1.728135697432896, 0, 0, 1.728135697432896, 1.986374781128185, 3.274347855143445, 3.924558228805985, 4.494675021618772, 5.086763598563966, 5.639636753851219, 6.560463682239176, 6.987916175323969, 7.609886518008811, 8.191645328076428, 8.721712191905006, 9.342000473839292, 10.10109620827608, 10.81678409657385, 11.09063938409442, 11.57023192490572, 12.20142037339500, 12.60203664300314, 13.27445027919837, 13.75243125781431, 14.73202675395121, 14.93715413369038, 15.31925932656467, 16.07151343376059

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