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-s − 3-s − 4-s + 2·5-s − 6-s + 4·7-s − 3·8-s + 9-s + 2·10-s + 12-s + 4·14-s − 2·15-s − 16-s + 2·17-s + 18-s − 2·20-s − 4·21-s + 8·23-s + 3·24-s − 25-s − 27-s − 4·28-s + 6·29-s − 2·30-s + 8·31-s + 5·32-s + 2·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s + 1.51·7-s − 1.06·8-s + 1/3·9-s + 0.632·10-s + 0.288·12-s + 1.06·14-s − 0.516·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.447·20-s − 0.872·21-s + 1.66·23-s + 0.612·24-s − 1/5·25-s − 0.192·27-s − 0.755·28-s + 1.11·29-s − 0.365·30-s + 1.43·31-s + 0.883·32-s + 0.342·34-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  =  $(2,\ 61347,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.131319888$
$L(\frac12)$  $\approx$  $4.131319888$
$L(\frac{3}{2})$   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 \( 1 - T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−14.15226877526794, −13.82120460010251, −13.34621560549036, −12.91543696417800, −12.20806740865185, −11.80485117401437, −11.48195680204925, −10.68319173629536, −10.31580024118681, −9.769213913506245, −9.143093324402864, −8.579247263212522, −8.191235288639512, −7.483791394112557, −6.727865898363266, −6.282863410104654, −5.560398743568963, −5.229533852702573, −4.719604316475137, −4.434358256951043, −3.469566004011246, −2.862026986356606, −2.041356170120436, −1.300814858812808, −0.7055607459001401, 0.7055607459001401, 1.300814858812808, 2.041356170120436, 2.862026986356606, 3.469566004011246, 4.434358256951043, 4.719604316475137, 5.229533852702573, 5.560398743568963, 6.282863410104654, 6.727865898363266, 7.483791394112557, 8.191235288639512, 8.579247263212522, 9.143093324402864, 9.769213913506245, 10.31580024118681, 10.68319173629536, 11.48195680204925, 11.80485117401437, 12.20806740865185, 12.91543696417800, 13.34621560549036, 13.82120460010251, 14.15226877526794

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