Properties

Label 2-9450-1.1-c1-0-137
Degree $2$
Conductor $9450$
Sign $-1$
Analytic cond. $75.4586$
Root an. cond. $8.68669$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 7-s + 8-s − 3·11-s − 5·13-s + 14-s + 16-s + 4·17-s + 19-s − 3·22-s + 8·23-s − 5·26-s + 28-s − 10·29-s − 8·31-s + 32-s + 4·34-s + 4·37-s + 38-s − 9·41-s − 43-s − 3·44-s + 8·46-s − 13·47-s + 49-s − 5·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 0.904·11-s − 1.38·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.229·19-s − 0.639·22-s + 1.66·23-s − 0.980·26-s + 0.188·28-s − 1.85·29-s − 1.43·31-s + 0.176·32-s + 0.685·34-s + 0.657·37-s + 0.162·38-s − 1.40·41-s − 0.152·43-s − 0.452·44-s + 1.17·46-s − 1.89·47-s + 1/7·49-s − 0.693·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9450\)    =    \(2 \cdot 3^{3} \cdot 5^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(75.4586\)
Root analytic conductor: \(8.68669\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9450,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
good11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 13 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−7.23020346724023295715235118343, −6.87780545880214748996879866598, −5.58629589844294514229000995054, −5.30631382946002431245555925664, −4.81352372083824645681716851393, −3.76890647058484597839516015603, −3.09515664551022474479065769615, −2.33335115239000221504526264039, −1.45018831225646045227457082136, 0, 1.45018831225646045227457082136, 2.33335115239000221504526264039, 3.09515664551022474479065769615, 3.76890647058484597839516015603, 4.81352372083824645681716851393, 5.30631382946002431245555925664, 5.58629589844294514229000995054, 6.87780545880214748996879866598, 7.23020346724023295715235118343

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