Properties

Degree 2
Conductor $ 3^{2} \cdot 5^{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  − 2·4-s + 5·7-s + 5·13-s + 4·16-s − 19-s − 10·28-s − 7·31-s − 10·37-s + 5·43-s + 18·49-s − 10·52-s − 13·61-s − 8·64-s + 5·67-s − 10·73-s + 2·76-s − 4·79-s + 25·91-s + 5·97-s + 20·103-s − 19·109-s + 20·112-s + ⋯
L(s)  = 1  − 4-s + 1.88·7-s + 1.38·13-s + 16-s − 0.229·19-s − 1.88·28-s − 1.25·31-s − 1.64·37-s + 0.762·43-s + 18/7·49-s − 1.38·52-s − 1.66·61-s − 64-s + 0.610·67-s − 1.17·73-s + 0.229·76-s − 0.450·79-s + 2.62·91-s + 0.507·97-s + 1.97·103-s − 1.81·109-s + 1.88·112-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  =  0
Selberg data  =  $(2,\ 225,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.20836$
$L(\frac12)$  $\approx$  $1.20836$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
7 \( 1 - 5 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−12.23814096251703440015939659085, −11.17123610633436781993835627791, −10.50317121206251201104711594578, −8.995186005196822049947024673068, −8.459962290175805266187907432817, −7.51612763708038261050643932552, −5.77556571014558049232659651532, −4.83022728068118968978622921639, −3.80273248963840130172861112356, −1.51470298744943699034589542418, 1.51470298744943699034589542418, 3.80273248963840130172861112356, 4.83022728068118968978622921639, 5.77556571014558049232659651532, 7.51612763708038261050643932552, 8.459962290175805266187907432817, 8.995186005196822049947024673068, 10.50317121206251201104711594578, 11.17123610633436781993835627791, 12.23814096251703440015939659085

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