Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 11^{2} $
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 + 2·7-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1452\)    =    \(2^{2} \cdot 3 \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1452} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 1452,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.750000211$
$L(\frac12)$  $\approx$  $2.750000211$
$L(\frac{3}{2})$   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

−19.62790879350697, −18.39039434019286, −18.14990975364372, −17.60068769053171, −16.71959427905468, −15.93432423396218, −15.32830634551997, −14.38759807287693, −14.03908363970586, −13.32917776412733, −12.75485562861439, −11.57393622664451, −11.13707506491211, −10.06310875004499, −9.555194722852196, −8.694963925814389, −8.069373725115527, −7.134584775437043, −6.253406077824870, −5.336489359657153, −4.503225133464154, −3.352476800870921, −2.267694675554005, −1.335757341482600, 1.335757341482600, 2.267694675554005, 3.352476800870921, 4.503225133464154, 5.336489359657153, 6.253406077824870, 7.134584775437043, 8.069373725115527, 8.694963925814389, 9.555194722852196, 10.06310875004499, 11.13707506491211, 11.57393622664451, 12.75485562861439, 13.32917776412733, 14.03908363970586, 14.38759807287693, 15.32830634551997, 15.93432423396218, 16.71959427905468, 17.60068769053171, 18.14990975364372, 18.39039434019286, 19.62790879350697

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