Properties

Degree 2
Conductor $ 3^{2} \cdot 5^{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  − 2·2-s + 2·4-s − 3·7-s − 2·11-s + 13-s + 6·14-s − 4·16-s − 2·17-s − 5·19-s + 4·22-s − 6·23-s − 2·26-s − 6·28-s − 10·29-s − 3·31-s + 8·32-s + 4·34-s + 2·37-s + 10·38-s + 8·41-s + 43-s − 4·44-s + 12·46-s − 2·47-s + 2·49-s + 2·52-s + 4·53-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 1.13·7-s − 0.603·11-s + 0.277·13-s + 1.60·14-s − 16-s − 0.485·17-s − 1.14·19-s + 0.852·22-s − 1.25·23-s − 0.392·26-s − 1.13·28-s − 1.85·29-s − 0.538·31-s + 1.41·32-s + 0.685·34-s + 0.328·37-s + 1.62·38-s + 1.24·41-s + 0.152·43-s − 0.603·44-s + 1.76·46-s − 0.291·47-s + 2/7·49-s + 0.277·52-s + 0.549·53-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(225\)    =    \(3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{225} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 225,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + p T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−11.36506854345797939837793722937, −10.51847852466115917559094220478, −9.725369907410972008022337048795, −8.938558667143995354006242020404, −7.965886472126962800720761127653, −6.96141726621690601574492077524, −5.87930303915993667052597404250, −4.01465426298836833468759021488, −2.23186561512245260821665501913, 0, 2.23186561512245260821665501913, 4.01465426298836833468759021488, 5.87930303915993667052597404250, 6.96141726621690601574492077524, 7.965886472126962800720761127653, 8.938558667143995354006242020404, 9.725369907410972008022337048795, 10.51847852466115917559094220478, 11.36506854345797939837793722937

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