Properties

Degree 2
Conductor $ 3 \cdot 5^{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 − 6-s + 3·8-s + 9-s + 4·11-s − 12-s − 16-s − 2·17-s − 18-s − 4·19-s − 4·22-s + 3·24-s + 27-s − 2·29-s − 5·32-s + 4·33-s + 2·34-s − 36-s − 10·37-s + 4·38-s − 10·41-s − 4·43-s − 4·44-s + 8·47-s − 48-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.06·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s − 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.852·22-s + 0.612·24-s + 0.192·27-s − 0.371·29-s − 0.883·32-s + 0.696·33-s + 0.342·34-s − 1/6·36-s − 1.64·37-s + 0.648·38-s − 1.56·41-s − 0.609·43-s − 0.603·44-s + 1.16·47-s − 0.144·48-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 12675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(12675\)    =    \(3 \cdot 5^{2} \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{12675} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 12675,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.389787412$
$L(\frac12)$  $\approx$  $1.389787412$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−16.54829543788129, −15.67760793329753, −15.15924464120678, −14.59122930274524, −13.97061073130418, −13.63168461932921, −12.96145751889923, −12.37528564052261, −11.68902764890869, −11.00152731462433, −10.30475672724044, −9.908406613822069, −9.086700852574739, −8.836899085764887, −8.330779704982344, −7.631393990594670, −6.820274632927206, −6.501097659061961, −5.316786989673608, −4.739135771892592, −3.818108723996313, −3.604810099796113, −2.208096203798496, −1.645507931283537, −0.5932000504268315, 0.5932000504268315, 1.645507931283537, 2.208096203798496, 3.604810099796113, 3.818108723996313, 4.739135771892592, 5.316786989673608, 6.501097659061961, 6.820274632927206, 7.631393990594670, 8.330779704982344, 8.836899085764887, 9.086700852574739, 9.908406613822069, 10.30475672724044, 11.00152731462433, 11.68902764890869, 12.37528564052261, 12.96145751889923, 13.63168461932921, 13.97061073130418, 14.59122930274524, 15.15924464120678, 15.67760793329753, 16.54829543788129

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