Properties

Degree 2
Conductor $ 3 \cdot 11^{2} \cdot 13^{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  + 2·2-s − 3-s + 2·4-s − 2·6-s + 9-s − 2·12-s − 4·16-s + 2·17-s + 2·18-s − 5·19-s − 6·23-s − 5·25-s − 27-s + 4·29-s − 8·32-s + 4·34-s + 2·36-s + 3·37-s − 10·38-s − 11·43-s − 12·46-s + 4·47-s + 4·48-s − 7·49-s − 10·50-s − 2·51-s − 12·53-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s + 1/3·9-s − 0.577·12-s − 16-s + 0.485·17-s + 0.471·18-s − 1.14·19-s − 1.25·23-s − 25-s − 0.192·27-s + 0.742·29-s − 1.41·32-s + 0.685·34-s + 1/3·36-s + 0.493·37-s − 1.62·38-s − 1.67·43-s − 1.76·46-s + 0.583·47-s + 0.577·48-s − 49-s − 1.41·50-s − 0.280·51-s − 1.64·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(61347\)    =    \(3 \cdot 11^{2} \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{61347} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 61347,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.005506634\)
\(L(\frac12)\)  \(\approx\)  \(2.005506634\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;11,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;11,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 - p T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−14.28361909784555, −13.76397784363055, −13.21373419429686, −12.85649014728295, −12.24378814994203, −11.97461799027370, −11.44168087375457, −10.96676372356048, −10.31013691170074, −9.808259131789362, −9.325578270225856, −8.337535736136572, −8.147167182984335, −7.284805407034447, −6.652899323739941, −6.133185475029755, −5.926748299416919, −5.111111840240153, −4.729260907797067, −4.103635271688309, −3.658805605921154, −2.969265443862830, −2.176950488389698, −1.614422908687145, −0.3705300212594226, 0.3705300212594226, 1.614422908687145, 2.176950488389698, 2.969265443862830, 3.658805605921154, 4.103635271688309, 4.729260907797067, 5.111111840240153, 5.926748299416919, 6.133185475029755, 6.652899323739941, 7.284805407034447, 8.147167182984335, 8.337535736136572, 9.325578270225856, 9.808259131789362, 10.31013691170074, 10.96676372356048, 11.44168087375457, 11.97461799027370, 12.24378814994203, 12.85649014728295, 13.21373419429686, 13.76397784363055, 14.28361909784555

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