↑

Properties

Label 2-3e2-1.1-c9-0-2
Degree $2$
Conductor $9$
Sign $-1$
Analytic cond. $4.63532$
Root an. cond. $2.15297$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 512·4-s − 1.25e4·7-s + 1.18e5·13-s + 2.62e5·16-s − 9.76e5·19-s − 1.95e6·25-s + 6.44e6·28-s + 1.69e6·31-s − 1.53e7·37-s − 1.65e7·43-s + 1.17e8·49-s − 6.06e7·52-s − 1.17e8·61-s − 1.34e8·64-s + 1.12e8·67-s + 2.96e8·73-s + 5.00e8·76-s − 6.16e8·79-s − 1.48e9·91-s + 1.28e9·97-s + 1.00e9·100-s + 6.22e7·103-s + 2.24e9·109-s − 3.29e9·112-s + ⋯
L(s)  = 1  − 4-s − 1.98·7-s + 1.14·13-s + 16-s − 1.71·19-s − 25-s + 1.98·28-s + 0.328·31-s − 1.34·37-s − 0.739·43-s + 2.92·49-s − 1.14·52-s − 1.09·61-s − 64-s + 0.682·67-s + 1.22·73-s + 1.71·76-s − 1.78·79-s − 2.27·91-s + 1.47·97-s + 100-s + 0.0545·103-s + 1.52·109-s − 1.98·112-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $-1$
Analytic conductor: \(4.63532\)
Root analytic conductor: \(2.15297\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + p^{9} T^{2} \)
5 \( 1 + p^{9} T^{2} \)
7 \( 1 + 12580 T + p^{9} T^{2} \)
11 \( 1 + p^{9} T^{2} \)
13 \( 1 - 118370 T + p^{9} T^{2} \)
17 \( 1 + p^{9} T^{2} \)
19 \( 1 + 976696 T + p^{9} T^{2} \)
23 \( 1 + p^{9} T^{2} \)
29 \( 1 + p^{9} T^{2} \)
31 \( 1 - 1691228 T + p^{9} T^{2} \)
37 \( 1 + 15384490 T + p^{9} T^{2} \)
41 \( 1 + p^{9} T^{2} \)
43 \( 1 + 16577080 T + p^{9} T^{2} \)
47 \( 1 + p^{9} T^{2} \)
53 \( 1 + p^{9} T^{2} \)
59 \( 1 + p^{9} T^{2} \)
61 \( 1 + 117903058 T + p^{9} T^{2} \)
67 \( 1 - 112542320 T + p^{9} T^{2} \)
71 \( 1 + p^{9} T^{2} \)
73 \( 1 - 296368310 T + p^{9} T^{2} \)
79 \( 1 + 616732324 T + p^{9} T^{2} \)
83 \( 1 + p^{9} T^{2} \)
89 \( 1 + p^{9} T^{2} \)
97 \( 1 - 1288928270 T + p^{9} T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.65771400408966928132105338978, −17.01229941409025196420308296020, −15.60791149521446041112886241683, −13.61520421093302631919623762008, −12.68709146287304402273599092082, −10.17140080936791085162008535274, −8.799319136821458946866051583895, −6.23790552157025490503551065115, −3.70593747015842403415754514185, 0, 3.70593747015842403415754514185, 6.23790552157025490503551065115, 8.799319136821458946866051583895, 10.17140080936791085162008535274, 12.68709146287304402273599092082, 13.61520421093302631919623762008, 15.60791149521446041112886241683, 17.01229941409025196420308296020, 18.65771400408966928132105338978

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