Properties

Label 2-9075-1.1-c1-0-61
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s − 1.73·7-s + 9-s − 2·12-s + 4·16-s + 3.46·17-s − 5.19·19-s − 1.73·21-s − 6·23-s + 27-s + 3.46·28-s + 6.92·29-s + 31-s − 2·36-s + 5·37-s + 3.46·41-s − 10.3·43-s + 12·47-s + 4·48-s − 4·49-s + 3.46·51-s − 6·53-s − 5.19·57-s − 12.1·61-s − 1.73·63-s − 8·64-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.654·7-s + 0.333·9-s − 0.577·12-s + 16-s + 0.840·17-s − 1.19·19-s − 0.377·21-s − 1.25·23-s + 0.192·27-s + 0.654·28-s + 1.28·29-s + 0.179·31-s − 0.333·36-s + 0.821·37-s + 0.541·41-s − 1.58·43-s + 1.75·47-s + 0.577·48-s − 0.571·49-s + 0.485·51-s − 0.824·53-s − 0.688·57-s − 1.55·61-s − 0.218·63-s − 64-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: \(0\)
Selberg data: \((2,\ 9075,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.436641876\)
\(L(\frac12)\) \(\approx\) \(1.436641876\)
\(L(\frac{3}{2})\) 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 \( 1 + 2T^{2} \)
7 \( 1 + 1.73T + 7T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 + 5.19T + 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 - 6.92T + 29T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 - 5T + 37T^{2} \)
41 \( 1 - 3.46T + 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 12.1T + 61T^{2} \)
67 \( 1 - 5T + 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 1.73T + 73T^{2} \)
79 \( 1 + 15.5T + 79T^{2} \)
83 \( 1 - 6.92T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 13T + 97T^{2} \)
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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

−8.032054782214698978927112473398, −7.13711717080999035317305935993, −6.22958245868092253509088806722, −5.80486623397408439772546422653, −4.69339965261509922777148182097, −4.27044203224783758105407073860, −3.45397851510919041401784544714, −2.81981699177738900059336227282, −1.72958462479312913654454533508, −0.56602574337960999418448541241, 0.56602574337960999418448541241, 1.72958462479312913654454533508, 2.81981699177738900059336227282, 3.45397851510919041401784544714, 4.27044203224783758105407073860, 4.69339965261509922777148182097, 5.80486623397408439772546422653, 6.22958245868092253509088806722, 7.13711717080999035317305935993, 8.032054782214698978927112473398

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