Properties

Label 2-9075-1.1-c1-0-262
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  + 0.618·2-s + 3-s − 1.61·4-s + 0.618·6-s − 0.763·7-s − 2.23·8-s + 9-s − 1.61·12-s − 1.23·13-s − 0.472·14-s + 1.85·16-s + 2·17-s + 0.618·18-s − 5·19-s − 0.763·21-s − 2.38·23-s − 2.23·24-s − 0.763·26-s + 27-s + 1.23·28-s + 5.85·29-s + 7·31-s + 5.61·32-s + 1.23·34-s − 1.61·36-s + 7.47·37-s − 3.09·38-s + ⋯
L(s)  = 1  + 0.437·2-s + 0.577·3-s − 0.809·4-s + 0.252·6-s − 0.288·7-s − 0.790·8-s + 0.333·9-s − 0.467·12-s − 0.342·13-s − 0.126·14-s + 0.463·16-s + 0.485·17-s + 0.145·18-s − 1.14·19-s − 0.166·21-s − 0.496·23-s − 0.456·24-s − 0.149·26-s + 0.192·27-s + 0.233·28-s + 1.08·29-s + 1.25·31-s + 0.993·32-s + 0.211·34-s − 0.269·36-s + 1.22·37-s − 0.501·38-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 \( 1 - 0.618T + 2T^{2} \)
7 \( 1 + 0.763T + 7T^{2} \)
13 \( 1 + 1.23T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + 2.38T + 23T^{2} \)
29 \( 1 - 5.85T + 29T^{2} \)
31 \( 1 - 7T + 31T^{2} \)
37 \( 1 - 7.47T + 37T^{2} \)
41 \( 1 + 2.52T + 41T^{2} \)
43 \( 1 + 2.09T + 43T^{2} \)
47 \( 1 + 5.09T + 47T^{2} \)
53 \( 1 - 7.61T + 53T^{2} \)
59 \( 1 + 6.70T + 59T^{2} \)
61 \( 1 + 7T + 61T^{2} \)
67 \( 1 - 9.38T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 13.4T + 73T^{2} \)
79 \( 1 - 8.09T + 79T^{2} \)
83 \( 1 + 5.70T + 83T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + 11.4T + 97T^{2} \)
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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

−7.54022256928087440928849863933, −6.45230954958800040088864973951, −6.15759014921972241762739739026, −5.09522940414689261137001707323, −4.52953916624470647373469999962, −3.93267732805240572372565729720, −3.08801349628430757443220575255, −2.48609794171826852903278564832, −1.23307826625401211238811328081, 0, 1.23307826625401211238811328081, 2.48609794171826852903278564832, 3.08801349628430757443220575255, 3.93267732805240572372565729720, 4.52953916624470647373469999962, 5.09522940414689261137001707323, 6.15759014921972241762739739026, 6.45230954958800040088864973951, 7.54022256928087440928849863933

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