Properties

Label 2-9075-1.1-c1-0-123
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 − 4.37·7-s + 1.73·8-s + 9-s + 1.79·12-s − 0.913·13-s + 1.99·14-s + 2.79·16-s − 1.73·17-s − 0.456·18-s − 3.46·19-s + 4.37·21-s + 0.582·23-s − 1.73·24-s + 0.417·26-s − 27-s + 7.84·28-s + 9.66·29-s − 8.58·31-s − 4.73·32-s + 0.791·34-s − 1.79·36-s − 7.58·37-s + 1.58·38-s + ⋯
L(s)  = 1  − 0.323·2-s − 0.577·3-s − 0.895·4-s + 0.186·6-s − 1.65·7-s + 0.612·8-s + 0.333·9-s + 0.517·12-s − 0.253·13-s + 0.534·14-s + 0.697·16-s − 0.420·17-s − 0.107·18-s − 0.794·19-s + 0.955·21-s + 0.121·23-s − 0.353·24-s + 0.0818·26-s − 0.192·27-s + 1.48·28-s + 1.79·29-s − 1.54·31-s − 0.837·32-s + 0.135·34-s − 0.298·36-s − 1.24·37-s + 0.256·38-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 \( 1 + 4.37T + 7T^{2} \)
13 \( 1 + 0.913T + 13T^{2} \)
17 \( 1 + 1.73T + 17T^{2} \)
19 \( 1 + 3.46T + 19T^{2} \)
23 \( 1 - 0.582T + 23T^{2} \)
29 \( 1 - 9.66T + 29T^{2} \)
31 \( 1 + 8.58T + 31T^{2} \)
37 \( 1 + 7.58T + 37T^{2} \)
41 \( 1 - 3.46T + 41T^{2} \)
43 \( 1 - 9.66T + 43T^{2} \)
47 \( 1 + 3.41T + 47T^{2} \)
53 \( 1 + 12.1T + 53T^{2} \)
59 \( 1 - 13.5T + 59T^{2} \)
61 \( 1 + 3.55T + 61T^{2} \)
67 \( 1 - 4.41T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 5.10T + 73T^{2} \)
79 \( 1 - 0.818T + 79T^{2} \)
83 \( 1 - 15.8T + 83T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 + 4.41T + 97T^{2} \)
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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

−7.27528955124402455408403814817, −6.66492195735715093893452352801, −6.11413865756060339747790658620, −5.32886240285770437887100405692, −4.61334101720168210885867718852, −3.87298226098242142882458249367, −3.20695816757599765876150856282, −2.13139145476713947561444506885, −0.789459359414424306136116193940, 0, 0.789459359414424306136116193940, 2.13139145476713947561444506885, 3.20695816757599765876150856282, 3.87298226098242142882458249367, 4.61334101720168210885867718852, 5.32886240285770437887100405692, 6.11413865756060339747790658620, 6.66492195735715093893452352801, 7.27528955124402455408403814817

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