Properties

Degree 2
Conductor $ 3 \cdot 7^{2} \cdot 11 \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 − 3·8-s + 9-s − 2·10-s + 11-s − 12-s − 2·15-s − 16-s − 2·17-s + 18-s + 4·19-s + 2·20-s + 22-s − 3·24-s − 25-s + 27-s − 2·29-s − 2·30-s + 8·31-s + 5·32-s + 33-s − 2·34-s − 36-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s − 0.288·12-s − 0.516·15-s − 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.447·20-s + 0.213·22-s − 0.612·24-s − 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.365·30-s + 1.43·31-s + 0.883·32-s + 0.174·33-s − 0.342·34-s − 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(273273\)    =    \(3 \cdot 7^{2} \cdot 11 \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{273273} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 273273,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.315677501$
$L(\frac12)$  $\approx$  $3.315677501$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 \)
11 \( 1 - T \)
13 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−12.89632097125228, −12.32048365476283, −11.83009097629736, −11.70574015520308, −11.06565910456552, −10.49616526988381, −9.799618134779705, −9.558935008094289, −9.078196377504634, −8.413005309063417, −8.248318404468871, −7.705795004383067, −7.138509172596178, −6.604709444701420, −6.175936560612405, −5.413461113386694, −5.037683208308491, −4.515521038212013, −3.977696050335610, −3.612072364580318, −3.236146627217582, −2.523768899499871, −1.985444563073240, −0.9563644894770993, −0.5108526991881828, 0.5108526991881828, 0.9563644894770993, 1.985444563073240, 2.523768899499871, 3.236146627217582, 3.612072364580318, 3.977696050335610, 4.515521038212013, 5.037683208308491, 5.413461113386694, 6.175936560612405, 6.604709444701420, 7.138509172596178, 7.705795004383067, 8.248318404468871, 8.413005309063417, 9.078196377504634, 9.558935008094289, 9.799618134779705, 10.49616526988381, 11.06565910456552, 11.70574015520308, 11.83009097629736, 12.32048365476283, 12.89632097125228

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