Properties

Label 2-9075-1.1-c1-0-113
Degree $2$
Conductor $9075$
Sign $1$
Analytic cond. $72.4642$
Root an. cond. $8.51259$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·4-s − 7-s + 9-s + 2·12-s + 2·13-s + 4·16-s + 6·17-s + 7·19-s + 21-s + 6·23-s − 27-s + 2·28-s − 31-s − 2·36-s + 7·37-s − 2·39-s + 6·41-s + 8·43-s − 4·48-s − 6·49-s − 6·51-s − 4·52-s + 6·53-s − 7·57-s − 12·59-s + 61-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 0.377·7-s + 1/3·9-s + 0.577·12-s + 0.554·13-s + 16-s + 1.45·17-s + 1.60·19-s + 0.218·21-s + 1.25·23-s − 0.192·27-s + 0.377·28-s − 0.179·31-s − 1/3·36-s + 1.15·37-s − 0.320·39-s + 0.937·41-s + 1.21·43-s − 0.577·48-s − 6/7·49-s − 0.840·51-s − 0.554·52-s + 0.824·53-s − 0.927·57-s − 1.56·59-s + 0.128·61-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: yes
Arithmetic: yes
Character: $\chi_{9075} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9075,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.541823506\)
\(L(\frac12)\) \(\approx\) \(1.541823506\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−7.57429178589206752599825058577, −7.26171901265395654422771855257, −6.04510595268873663188530754064, −5.71649178853268531344450376349, −5.02152432805084214325173207341, −4.31963574436031364627776411170, −3.45739773570944288331043111385, −2.93093200895283056957551589461, −1.26269753115414327598647616864, −0.74876154913487424076601045851, 0.74876154913487424076601045851, 1.26269753115414327598647616864, 2.93093200895283056957551589461, 3.45739773570944288331043111385, 4.31963574436031364627776411170, 5.02152432805084214325173207341, 5.71649178853268531344450376349, 6.04510595268873663188530754064, 7.26171901265395654422771855257, 7.57429178589206752599825058577

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