Properties

Degree 2
Conductor $ 3 \cdot 11^{2} \cdot 13^{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  + 3-s − 2·4-s − 7-s + 9-s − 2·12-s + 4·16-s − 6·17-s − 19-s − 21-s + 6·23-s − 5·25-s + 27-s + 2·28-s + 7·31-s − 2·36-s − 5·37-s + 4·43-s + 6·47-s + 4·48-s − 6·49-s − 6·51-s − 57-s + 6·59-s + 61-s − 63-s − 8·64-s − 5·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.377·7-s + 1/3·9-s − 0.577·12-s + 16-s − 1.45·17-s − 0.229·19-s − 0.218·21-s + 1.25·23-s − 25-s + 0.192·27-s + 0.377·28-s + 1.25·31-s − 1/3·36-s − 0.821·37-s + 0.609·43-s + 0.875·47-s + 0.577·48-s − 6/7·49-s − 0.840·51-s − 0.132·57-s + 0.781·59-s + 0.128·61-s − 0.125·63-s − 64-s − 0.610·67-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  =  \(1\)
Selberg data  =  \((2,\ 61347,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;11,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]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 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−14.46997283353544, −13.85953750252588, −13.58619586865843, −13.14555254600258, −12.70649151079161, −12.16005655008649, −11.53127216290532, −10.85597762701478, −10.41474722301332, −9.780079197255395, −9.337024702869771, −8.923532151946102, −8.444318936541381, −7.976422238763618, −7.286948181865011, −6.695625589657726, −6.206366359102985, −5.391247077346834, −4.898005969591190, −4.199821564264871, −3.918336427996354, −3.074995033757832, −2.559996399987458, −1.729854172585306, −0.8436685276696063, 0, 0.8436685276696063, 1.729854172585306, 2.559996399987458, 3.074995033757832, 3.918336427996354, 4.199821564264871, 4.898005969591190, 5.391247077346834, 6.206366359102985, 6.695625589657726, 7.286948181865011, 7.976422238763618, 8.444318936541381, 8.923532151946102, 9.337024702869771, 9.780079197255395, 10.41474722301332, 10.85597762701478, 11.53127216290532, 12.16005655008649, 12.70649151079161, 13.14555254600258, 13.58619586865843, 13.85953750252588, 14.46997283353544

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