Properties

Label 2-9072-1.1-c1-0-87
Degree $2$
Conductor $9072$
Sign $-1$
Analytic cond. $72.4402$
Root an. cond. $8.51118$
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  − 2.53·5-s − 7-s + 0.467·11-s + 5.82·13-s − 3.87·17-s + 2.18·19-s − 0.106·23-s + 1.41·25-s − 8.78·29-s + 7.68·31-s + 2.53·35-s − 7.68·37-s + 2.22·41-s − 1.22·43-s − 5.33·47-s + 49-s + 0.716·53-s − 1.18·55-s + 0.736·59-s + 0.958·61-s − 14.7·65-s + 9.63·67-s + 13.2·71-s − 10.2·73-s − 0.467·77-s + 12.6·79-s − 2.73·83-s + ⋯
L(s)  = 1  − 1.13·5-s − 0.377·7-s + 0.141·11-s + 1.61·13-s − 0.940·17-s + 0.501·19-s − 0.0221·23-s + 0.282·25-s − 1.63·29-s + 1.37·31-s + 0.428·35-s − 1.26·37-s + 0.347·41-s − 0.187·43-s − 0.777·47-s + 0.142·49-s + 0.0984·53-s − 0.159·55-s + 0.0958·59-s + 0.122·61-s − 1.82·65-s + 1.17·67-s + 1.57·71-s − 1.20·73-s − 0.0533·77-s + 1.42·79-s − 0.299·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9072\)    =    \(2^{4} \cdot 3^{4} \cdot 7\)
Sign: $-1$
Analytic conductor: \(72.4402\)
Root analytic conductor: \(8.51118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9072,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 + 2.53T + 5T^{2} \)
11 \( 1 - 0.467T + 11T^{2} \)
13 \( 1 - 5.82T + 13T^{2} \)
17 \( 1 + 3.87T + 17T^{2} \)
19 \( 1 - 2.18T + 19T^{2} \)
23 \( 1 + 0.106T + 23T^{2} \)
29 \( 1 + 8.78T + 29T^{2} \)
31 \( 1 - 7.68T + 31T^{2} \)
37 \( 1 + 7.68T + 37T^{2} \)
41 \( 1 - 2.22T + 41T^{2} \)
43 \( 1 + 1.22T + 43T^{2} \)
47 \( 1 + 5.33T + 47T^{2} \)
53 \( 1 - 0.716T + 53T^{2} \)
59 \( 1 - 0.736T + 59T^{2} \)
61 \( 1 - 0.958T + 61T^{2} \)
67 \( 1 - 9.63T + 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 - 12.6T + 79T^{2} \)
83 \( 1 + 2.73T + 83T^{2} \)
89 \( 1 - 8.11T + 89T^{2} \)
97 \( 1 + 13.6T + 97T^{2} \)
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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

−7.39461633405108169721374695461, −6.71499005643958701631963020505, −6.15580742549110452741072774481, −5.31190648056582993354416313286, −4.43675889862898862476441443736, −3.68490593594605929558762716827, −3.40819644287094852944877013445, −2.19580330199352117706180210153, −1.10062158697882348182059365399, 0, 1.10062158697882348182059365399, 2.19580330199352117706180210153, 3.40819644287094852944877013445, 3.68490593594605929558762716827, 4.43675889862898862476441443736, 5.31190648056582993354416313286, 6.15580742549110452741072774481, 6.71499005643958701631963020505, 7.39461633405108169721374695461

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