Properties

Label 2-3e2-1.1-c3-0-0
Degree $2$
Conductor $9$
Sign $1$
Analytic cond. $0.531017$
Root an. cond. $0.728709$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·4-s + 20·7-s − 70·13-s + 64·16-s + 56·19-s − 125·25-s − 160·28-s + 308·31-s + 110·37-s − 520·43-s + 57·49-s + 560·52-s + 182·61-s − 512·64-s − 880·67-s + 1.19e3·73-s − 448·76-s + 884·79-s − 1.40e3·91-s − 1.33e3·97-s + 1.00e3·100-s + 1.82e3·103-s − 646·109-s + 1.28e3·112-s + ⋯
L(s)  = 1  − 4-s + 1.07·7-s − 1.49·13-s + 16-s + 0.676·19-s − 25-s − 1.07·28-s + 1.78·31-s + 0.488·37-s − 1.84·43-s + 0.166·49-s + 1.49·52-s + 0.382·61-s − 64-s − 1.60·67-s + 1.90·73-s − 0.676·76-s + 1.25·79-s − 1.61·91-s − 1.39·97-s + 100-s + 1.74·103-s − 0.567·109-s + 1.07·112-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $1$
Analytic conductor: \(0.531017\)
Root analytic conductor: \(0.728709\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{9} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7599657499\)
\(L(\frac12)\) \(\approx\) \(0.7599657499\)
\(L(\frac{5}{2})\) 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 \( 1 + p^{3} T^{2} \)
5 \( 1 + p^{3} T^{2} \)
7 \( 1 - 20 T + p^{3} T^{2} \)
11 \( 1 + p^{3} T^{2} \)
13 \( 1 + 70 T + p^{3} T^{2} \)
17 \( 1 + p^{3} T^{2} \)
19 \( 1 - 56 T + p^{3} T^{2} \)
23 \( 1 + p^{3} T^{2} \)
29 \( 1 + p^{3} T^{2} \)
31 \( 1 - 308 T + p^{3} T^{2} \)
37 \( 1 - 110 T + p^{3} T^{2} \)
41 \( 1 + p^{3} T^{2} \)
43 \( 1 + 520 T + p^{3} T^{2} \)
47 \( 1 + p^{3} T^{2} \)
53 \( 1 + p^{3} T^{2} \)
59 \( 1 + p^{3} T^{2} \)
61 \( 1 - 182 T + p^{3} T^{2} \)
67 \( 1 + 880 T + p^{3} T^{2} \)
71 \( 1 + p^{3} T^{2} \)
73 \( 1 - 1190 T + p^{3} T^{2} \)
79 \( 1 - 884 T + p^{3} T^{2} \)
83 \( 1 + p^{3} T^{2} \)
89 \( 1 + p^{3} T^{2} \)
97 \( 1 + 1330 T + p^{3} T^{2} \)
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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

−21.24661792052083109285108792256, −19.55575161404065204005050768737, −18.07795890634270969575136593189, −17.12497617568479204935496377782, −14.93321934083369061780725752962, −13.74456961426938286092483499795, −11.94315109526654413183623103861, −9.820987617456067807528942581344, −7.996008136155744588027072213229, −4.88396746546602162317969751078, 4.88396746546602162317969751078, 7.996008136155744588027072213229, 9.820987617456067807528942581344, 11.94315109526654413183623103861, 13.74456961426938286092483499795, 14.93321934083369061780725752962, 17.12497617568479204935496377782, 18.07795890634270969575136593189, 19.55575161404065204005050768737, 21.24661792052083109285108792256

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