Properties

Label 2-9075-1.1-c1-0-332
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3-s + 2·4-s + 2·6-s − 7-s + 9-s + 2·12-s + 2·13-s − 2·14-s − 4·16-s − 4·17-s + 2·18-s − 3·19-s − 21-s − 2·23-s + 4·26-s + 27-s − 2·28-s + 6·29-s − 5·31-s − 8·32-s − 8·34-s + 2·36-s − 3·37-s − 6·38-s + 2·39-s − 2·41-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s − 0.377·7-s + 1/3·9-s + 0.577·12-s + 0.554·13-s − 0.534·14-s − 16-s − 0.970·17-s + 0.471·18-s − 0.688·19-s − 0.218·21-s − 0.417·23-s + 0.784·26-s + 0.192·27-s − 0.377·28-s + 1.11·29-s − 0.898·31-s − 1.41·32-s − 1.37·34-s + 1/3·36-s − 0.493·37-s − 0.973·38-s + 0.320·39-s − 0.312·41-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: 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 \( 1 - p T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−7.08086936620978505773832003698, −6.45579021166598527751933591708, −6.13999298258592868072826777601, −5.02168440562925838023081168891, −4.64893490109386035161986570280, −3.71473548580302558221557977493, −3.36614915710317352667973504359, −2.45828418333403566307465264333, −1.70851422044768225217083061731, 0, 1.70851422044768225217083061731, 2.45828418333403566307465264333, 3.36614915710317352667973504359, 3.71473548580302558221557977493, 4.64893490109386035161986570280, 5.02168440562925838023081168891, 6.13999298258592868072826777601, 6.45579021166598527751933591708, 7.08086936620978505773832003698

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