Properties

Degree 2
Conductor $ 3^{2} \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 − 4-s − 3·8-s − 4·11-s − 16-s + 2·17-s − 4·19-s − 4·22-s + 2·29-s + 5·32-s + 2·34-s − 10·37-s − 4·38-s + 10·41-s − 4·43-s + 4·44-s − 8·47-s − 7·49-s − 10·53-s + 2·58-s − 4·59-s − 2·61-s + 7·64-s + 12·67-s − 2·68-s − 8·71-s + 10·73-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s − 1.20·11-s − 1/4·16-s + 0.485·17-s − 0.917·19-s − 0.852·22-s + 0.371·29-s + 0.883·32-s + 0.342·34-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 − 49-s − 1.37·53-s + 0.262·58-s − 0.520·59-s − 0.256·61-s + 7/8·64-s + 1.46·67-s − 0.242·68-s − 0.949·71-s + 1.17·73-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(38025\)    =    \(3^{2} \cdot 5^{2} \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{38025} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 38025,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9144742382$
$L(\frac12)$  $\approx$  $0.9144742382$
$L(\frac{3}{2})$   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 \( 1 \)
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

−14.60739697870665, −14.38850645149216, −13.79401305822607, −13.20543771007693, −12.85930146219388, −12.39978505114474, −11.95755657371995, −11.04258370204017, −10.82170228675921, −9.945231596659135, −9.703424120049263, −8.945275556481718, −8.225714697200211, −8.112433309641856, −7.219685386680302, −6.527312553762123, −5.980103712600091, −5.349226168081437, −4.880321890804477, −4.391654426292730, −3.574710518392486, −3.082707304674841, −2.403870979514865, −1.496796016303638, −0.3092222087661859, 0.3092222087661859, 1.496796016303638, 2.403870979514865, 3.082707304674841, 3.574710518392486, 4.391654426292730, 4.880321890804477, 5.349226168081437, 5.980103712600091, 6.527312553762123, 7.219685386680302, 8.112433309641856, 8.225714697200211, 8.945275556481718, 9.703424120049263, 9.945231596659135, 10.82170228675921, 11.04258370204017, 11.95755657371995, 12.39978505114474, 12.85930146219388, 13.20543771007693, 13.79401305822607, 14.38850645149216, 14.60739697870665

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