Properties

Label 2-9075-1.1-c1-0-282
Degree $2$
Conductor $9075$
Sign $-1$
Analytic cond. $72.4642$
Root an. cond. $8.51259$
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  − 0.456·2-s + 3-s − 1.79·4-s − 0.456·6-s + 2.18·7-s + 1.73·8-s + 9-s − 1.79·12-s + 0.456·13-s − 0.999·14-s + 2.79·16-s − 1.73·17-s − 0.456·18-s − 1.73·19-s + 2.18·21-s − 1.20·23-s + 1.73·24-s − 0.208·26-s + 27-s − 3.92·28-s − 1.73·29-s − 3.20·31-s − 4.73·32-s + 0.791·34-s − 1.79·36-s + 1.58·37-s + 0.791·38-s + ⋯
L(s)  = 1  − 0.323·2-s + 0.577·3-s − 0.895·4-s − 0.186·6-s + 0.827·7-s + 0.612·8-s + 0.333·9-s − 0.517·12-s + 0.126·13-s − 0.267·14-s + 0.697·16-s − 0.420·17-s − 0.107·18-s − 0.397·19-s + 0.477·21-s − 0.252·23-s + 0.353·24-s − 0.0409·26-s + 0.192·27-s − 0.740·28-s − 0.321·29-s − 0.576·31-s − 0.837·32-s + 0.135·34-s − 0.298·36-s + 0.260·37-s + 0.128·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9075\)    =    \(3 \cdot 5^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(72.4642\)
Root analytic conductor: \(8.51259\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9075,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 - T \)
5 \( 1 \)
11 \( 1 \)
good2 \( 1 + 0.456T + 2T^{2} \)
7 \( 1 - 2.18T + 7T^{2} \)
13 \( 1 - 0.456T + 13T^{2} \)
17 \( 1 + 1.73T + 17T^{2} \)
19 \( 1 + 1.73T + 19T^{2} \)
23 \( 1 + 1.20T + 23T^{2} \)
29 \( 1 + 1.73T + 29T^{2} \)
31 \( 1 + 3.20T + 31T^{2} \)
37 \( 1 - 1.58T + 37T^{2} \)
41 \( 1 - 3.10T + 41T^{2} \)
43 \( 1 + 3.46T + 43T^{2} \)
47 \( 1 + 5.58T + 47T^{2} \)
53 \( 1 - 6.79T + 53T^{2} \)
59 \( 1 + 12.1T + 59T^{2} \)
61 \( 1 + 7.11T + 61T^{2} \)
67 \( 1 + 9.16T + 67T^{2} \)
71 \( 1 - 15.7T + 71T^{2} \)
73 \( 1 + 9.21T + 73T^{2} \)
79 \( 1 - 14.7T + 79T^{2} \)
83 \( 1 + 14.5T + 83T^{2} \)
89 \( 1 + 2.62T + 89T^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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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

−7.72384517685486898922119575174, −6.88203165222766742421299944909, −5.97355932753188294711944949267, −5.15683717530262001869640541700, −4.51951014040509095964014913441, −3.96518040619997374849666672399, −3.09738470770836203683167219012, −2.01292914680057450407109181792, −1.30235296757252558581240705994, 0, 1.30235296757252558581240705994, 2.01292914680057450407109181792, 3.09738470770836203683167219012, 3.96518040619997374849666672399, 4.51951014040509095964014913441, 5.15683717530262001869640541700, 5.97355932753188294711944949267, 6.88203165222766742421299944909, 7.72384517685486898922119575174

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