Properties

Label 2-9450-1.1-c1-0-94
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 + 11-s − 5·13-s + 14-s + 16-s + 4·17-s + 7·19-s − 22-s + 5·26-s − 28-s − 6·29-s − 32-s − 4·34-s + 4·37-s − 7·38-s − 3·41-s − 11·43-s + 44-s − 47-s + 49-s − 5·52-s − 53-s + 56-s + 6·58-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 0.301·11-s − 1.38·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s + 1.60·19-s − 0.213·22-s + 0.980·26-s − 0.188·28-s − 1.11·29-s − 0.176·32-s − 0.685·34-s + 0.657·37-s − 1.13·38-s − 0.468·41-s − 1.67·43-s + 0.150·44-s − 0.145·47-s + 1/7·49-s − 0.693·52-s − 0.137·53-s + 0.133·56-s + 0.787·58-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 \( 1 - T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 15 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−7.50160592426370673174582283932, −6.90257380824693566677364478275, −6.05974944204480736250084094484, −5.35994783068541243324439365136, −4.72299981886840165833391720298, −3.52205837382236619190595238157, −3.05657132792341723295549228367, −2.05744545230315636611377975368, −1.12957625216972102110433179538, 0, 1.12957625216972102110433179538, 2.05744545230315636611377975368, 3.05657132792341723295549228367, 3.52205837382236619190595238157, 4.72299981886840165833391720298, 5.35994783068541243324439365136, 6.05974944204480736250084094484, 6.90257380824693566677364478275, 7.50160592426370673174582283932

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