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  − 3-s − 2·4-s − 2·5-s + 5·7-s + 9-s + 2·12-s + 2·15-s + 4·16-s + 2·17-s + 19-s + 4·20-s − 5·21-s − 8·23-s − 25-s − 27-s − 10·28-s + 10·29-s + 3·31-s − 10·35-s − 2·36-s + 11·37-s − 4·41-s − 4·43-s − 2·45-s + 10·47-s − 4·48-s + 18·49-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 0.894·5-s + 1.88·7-s + 1/3·9-s + 0.577·12-s + 0.516·15-s + 16-s + 0.485·17-s + 0.229·19-s + 0.894·20-s − 1.09·21-s − 1.66·23-s − 1/5·25-s − 0.192·27-s − 1.88·28-s + 1.85·29-s + 0.538·31-s − 1.69·35-s − 1/3·36-s + 1.80·37-s − 0.624·41-s − 0.609·43-s − 0.298·45-s + 1.45·47-s − 0.577·48-s + 18/7·49-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  =  \((2,\ 61347,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.691087047\)
\(L(\frac12)\)  \(\approx\)  \(1.691087047\)
\(L(\frac{3}{2})\)   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 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 11 T + p T^{2} \)
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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

−14.23185996102122, −13.79689203757907, −13.52961550274912, −12.47327254787999, −12.20509230494066, −11.80233003666095, −11.35671078700277, −10.83494559799204, −10.08099840348392, −9.953851077836240, −9.025951367188670, −8.350019477807470, −8.107216192108693, −7.779139282142070, −7.165797569915637, −6.224476709343907, −5.719404504530667, −5.098860783010032, −4.593664261681408, −4.241407662416162, −3.774630203963918, −2.793044511114962, −1.889820243983507, −1.088185413845908, −0.5666767649650063, 0.5666767649650063, 1.088185413845908, 1.889820243983507, 2.793044511114962, 3.774630203963918, 4.241407662416162, 4.593664261681408, 5.098860783010032, 5.719404504530667, 6.224476709343907, 7.165797569915637, 7.779139282142070, 8.107216192108693, 8.350019477807470, 9.025951367188670, 9.953851077836240, 10.08099840348392, 10.83494559799204, 11.35671078700277, 11.80233003666095, 12.20509230494066, 12.47327254787999, 13.52961550274912, 13.79689203757907, 14.23185996102122

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