Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 11 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 2·7-s − 11-s − 2·13-s − 4·17-s − 6·19-s − 25-s + 8·29-s − 8·31-s + 4·35-s + 10·37-s − 8·41-s − 2·43-s + 8·47-s − 3·49-s + 2·53-s + 2·55-s − 12·59-s + 10·61-s + 4·65-s + 12·67-s − 8·71-s + 6·73-s + 2·77-s − 2·79-s − 16·83-s + 8·85-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.755·7-s − 0.301·11-s − 0.554·13-s − 0.970·17-s − 1.37·19-s − 1/5·25-s + 1.48·29-s − 1.43·31-s + 0.676·35-s + 1.64·37-s − 1.24·41-s − 0.304·43-s + 1.16·47-s − 3/7·49-s + 0.274·53-s + 0.269·55-s − 1.56·59-s + 1.28·61-s + 0.496·65-s + 1.46·67-s − 0.949·71-s + 0.702·73-s + 0.227·77-s − 0.225·79-s − 1.75·83-s + 0.867·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(396\)    =    \(2^{2} \cdot 3^{2} \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{396} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 396,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 \)
11 \( 1 + T \)
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

−19.85914980577309, −19.10321766313267, −18.38767368073400, −17.37411240373410, −16.61210849411450, −15.76458586274665, −15.25487968687179, −14.36173105015953, −13.19628901093509, −12.65067008103089, −11.73232994471417, −10.86740863890262, −9.989427953048951, −8.932199088171823, −8.071554028904628, −7.054307551808801, −6.200979152335185, −4.751187103460562, −3.798149485268459, −2.464836161396002, 0, 2.464836161396002, 3.798149485268459, 4.751187103460562, 6.200979152335185, 7.054307551808801, 8.071554028904628, 8.932199088171823, 9.989427953048951, 10.86740863890262, 11.73232994471417, 12.65067008103089, 13.19628901093509, 14.36173105015953, 15.25487968687179, 15.76458586274665, 16.61210849411450, 17.37411240373410, 18.38767368073400, 19.10321766313267, 19.85914980577309

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