Properties

Label 2-9450-1.1-c1-0-86
Degree $2$
Conductor $9450$
Sign $-1$
Analytic cond. $75.4586$
Root an. cond. $8.68669$
Motivic weight $1$
Arithmetic yes
Rational no
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 − 5.29·11-s − 0.645·13-s + 14-s + 16-s + 17-s + 4.29·19-s + 5.29·22-s + 3.64·23-s + 0.645·26-s − 28-s − 5.35·29-s + 3.29·31-s − 32-s − 34-s + 4.35·37-s − 4.29·38-s − 3.29·41-s − 5.64·43-s − 5.29·44-s − 3.64·46-s + 4.64·47-s + 49-s − 0.645·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.377·7-s − 0.353·8-s − 1.59·11-s − 0.179·13-s + 0.267·14-s + 0.250·16-s + 0.242·17-s + 0.984·19-s + 1.12·22-s + 0.760·23-s + 0.126·26-s − 0.188·28-s − 0.994·29-s + 0.591·31-s − 0.176·32-s − 0.171·34-s + 0.715·37-s − 0.696·38-s − 0.514·41-s − 0.860·43-s − 0.797·44-s − 0.537·46-s + 0.677·47-s + 0.142·49-s − 0.0895·52-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: no
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 \( 1 + 5.29T + 11T^{2} \)
13 \( 1 + 0.645T + 13T^{2} \)
17 \( 1 - T + 17T^{2} \)
19 \( 1 - 4.29T + 19T^{2} \)
23 \( 1 - 3.64T + 23T^{2} \)
29 \( 1 + 5.35T + 29T^{2} \)
31 \( 1 - 3.29T + 31T^{2} \)
37 \( 1 - 4.35T + 37T^{2} \)
41 \( 1 + 3.29T + 41T^{2} \)
43 \( 1 + 5.64T + 43T^{2} \)
47 \( 1 - 4.64T + 47T^{2} \)
53 \( 1 - 2.64T + 53T^{2} \)
59 \( 1 + 2.70T + 59T^{2} \)
61 \( 1 + 2.64T + 61T^{2} \)
67 \( 1 - 10.9T + 67T^{2} \)
71 \( 1 + 0.708T + 71T^{2} \)
73 \( 1 + 5.29T + 73T^{2} \)
79 \( 1 + 0.0627T + 79T^{2} \)
83 \( 1 - 1.64T + 83T^{2} \)
89 \( 1 + T + 89T^{2} \)
97 \( 1 + 2.93T + 97T^{2} \)
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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

−7.49329799301234944739690438689, −6.90478896529621312141069860609, −5.99967411579618404485099634653, −5.36988903429113802946218590036, −4.76434793954581180660242228266, −3.56364745157746435020668959124, −2.90382709360988810352616176132, −2.21357485607849914560923789968, −1.04546294330198868463872502981, 0, 1.04546294330198868463872502981, 2.21357485607849914560923789968, 2.90382709360988810352616176132, 3.56364745157746435020668959124, 4.76434793954581180660242228266, 5.36988903429113802946218590036, 5.99967411579618404485099634653, 6.90478896529621312141069860609, 7.49329799301234944739690438689

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