Properties

Label 2-15e2-1.1-c11-0-40
Degree $2$
Conductor $225$
Sign $1$
Analytic cond. $172.877$
Root an. cond. $13.1482$
Motivic weight $11$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.04e3·4-s + 7.68e4·7-s + 2.24e6·13-s + 4.19e6·16-s + 2.05e7·19-s − 1.57e8·28-s + 2.96e8·31-s + 7.82e8·37-s − 1.76e9·43-s + 3.93e9·49-s − 4.60e9·52-s − 1.29e10·61-s − 8.58e9·64-s − 5.95e9·67-s − 1.98e10·73-s − 4.21e10·76-s + 3.28e10·79-s + 1.72e11·91-s + 5.29e10·97-s + 7.30e10·103-s + 2.57e9·109-s + 3.22e11·112-s + ⋯
L(s)  = 1  − 4-s + 1.72·7-s + 1.67·13-s + 16-s + 1.90·19-s − 1.72·28-s + 1.85·31-s + 1.85·37-s − 1.83·43-s + 1.98·49-s − 1.67·52-s − 1.96·61-s − 64-s − 0.538·67-s − 1.11·73-s − 1.90·76-s + 1.20·79-s + 2.90·91-s + 0.626·97-s + 0.621·103-s + 0.0160·109-s + 1.72·112-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(172.877\)
Root analytic conductor: \(13.1482\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(3.048476299\)
\(L(\frac12)\) \(\approx\) \(3.048476299\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + p^{11} T^{2} \)
7 \( 1 - 76885 T + p^{11} T^{2} \)
11 \( 1 + p^{11} T^{2} \)
13 \( 1 - 2248615 T + p^{11} T^{2} \)
17 \( 1 + p^{11} T^{2} \)
19 \( 1 - 20581901 T + p^{11} T^{2} \)
23 \( 1 + p^{11} T^{2} \)
29 \( 1 + p^{11} T^{2} \)
31 \( 1 - 296476943 T + p^{11} T^{2} \)
37 \( 1 - 782919730 T + p^{11} T^{2} \)
41 \( 1 + p^{11} T^{2} \)
43 \( 1 + 1768358135 T + p^{11} T^{2} \)
47 \( 1 + p^{11} T^{2} \)
53 \( 1 + p^{11} T^{2} \)
59 \( 1 + p^{11} T^{2} \)
61 \( 1 + 12977292913 T + p^{11} T^{2} \)
67 \( 1 + 5951291615 T + p^{11} T^{2} \)
71 \( 1 + p^{11} T^{2} \)
73 \( 1 + 19805520230 T + p^{11} T^{2} \)
79 \( 1 - 32885832404 T + p^{11} T^{2} \)
83 \( 1 + p^{11} T^{2} \)
89 \( 1 + p^{11} T^{2} \)
97 \( 1 - 52968566635 T + p^{11} T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.26491711545580220947072577454, −9.166604036369049527349549965428, −8.268024259327521838157128765786, −7.74928044759126519839130330820, −6.06493277775770896823115596419, −5.06364737765706353861043761946, −4.33225916078312175025359751523, −3.16741634921039139068987136933, −1.42224706033779518287114174495, −0.904864232534800984867268570062, 0.904864232534800984867268570062, 1.42224706033779518287114174495, 3.16741634921039139068987136933, 4.33225916078312175025359751523, 5.06364737765706353861043761946, 6.06493277775770896823115596419, 7.74928044759126519839130330820, 8.268024259327521838157128765786, 9.166604036369049527349549965428, 10.26491711545580220947072577454

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