Properties

Degree 2
Conductor $ 2^{4} \cdot 5^{2} \cdot 11^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·7-s + 9-s + 2·13-s − 6·17-s − 4·19-s + 4·21-s + 6·23-s + 4·27-s − 6·29-s + 4·31-s − 2·37-s − 4·39-s − 6·41-s + 10·43-s − 6·47-s − 3·49-s + 12·51-s + 6·53-s + 8·57-s − 12·59-s − 2·61-s − 2·63-s + 2·67-s − 12·69-s + 12·71-s + 2·73-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.755·7-s + 1/3·9-s + 0.554·13-s − 1.45·17-s − 0.917·19-s + 0.872·21-s + 1.25·23-s + 0.769·27-s − 1.11·29-s + 0.718·31-s − 0.328·37-s − 0.640·39-s − 0.937·41-s + 1.52·43-s − 0.875·47-s − 3/7·49-s + 1.68·51-s + 0.824·53-s + 1.05·57-s − 1.56·59-s − 0.256·61-s − 0.251·63-s + 0.244·67-s − 1.44·69-s + 1.42·71-s + 0.234·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(48400\)    =    \(2^{4} \cdot 5^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{48400} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 48400,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−15.07203575106108, −14.22305534979671, −13.66938790674778, −13.11415415367032, −12.75681366346292, −12.33067500036664, −11.57231462383795, −11.14348395733178, −10.85332322775658, −10.35242442559050, −9.621785656272199, −9.050815530976243, −8.666685404653628, −7.981052292874027, −7.106704056644714, −6.666238366752031, −6.304924039475220, −5.791010533570492, −5.075436692668646, −4.600031327740585, −3.924777922731681, −3.206796656076863, −2.492014831537231, −1.670459424524384, −0.6901823178234976, 0, 0.6901823178234976, 1.670459424524384, 2.492014831537231, 3.206796656076863, 3.924777922731681, 4.600031327740585, 5.075436692668646, 5.791010533570492, 6.304924039475220, 6.666238366752031, 7.106704056644714, 7.981052292874027, 8.666685404653628, 9.050815530976243, 9.621785656272199, 10.35242442559050, 10.85332322775658, 11.14348395733178, 11.57231462383795, 12.33067500036664, 12.75681366346292, 13.11415415367032, 13.66938790674778, 14.22305534979671, 15.07203575106108

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