Properties

Label 2-9072-1.1-c1-0-91
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  − 1.34·5-s − 7-s + 1.65·11-s − 3.36·13-s − 0.467·17-s + 3.22·19-s + 8.94·23-s − 3.18·25-s − 6.26·29-s − 9.23·31-s + 1.34·35-s + 9.23·37-s − 3.41·41-s + 4.41·43-s + 9.35·47-s + 49-s + 0.573·53-s − 2.22·55-s − 10.3·59-s + 7.63·61-s + 4.53·65-s − 0.596·67-s − 0.554·71-s + 2.04·73-s − 1.65·77-s + 2.40·79-s − 15.0·83-s + ⋯
L(s)  = 1  − 0.602·5-s − 0.377·7-s + 0.498·11-s − 0.934·13-s − 0.113·17-s + 0.740·19-s + 1.86·23-s − 0.636·25-s − 1.16·29-s − 1.65·31-s + 0.227·35-s + 1.51·37-s − 0.532·41-s + 0.672·43-s + 1.36·47-s + 0.142·49-s + 0.0788·53-s − 0.300·55-s − 1.35·59-s + 0.977·61-s + 0.563·65-s − 0.0728·67-s − 0.0657·71-s + 0.239·73-s − 0.188·77-s + 0.270·79-s − 1.65·83-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 \( 1 + 1.34T + 5T^{2} \)
11 \( 1 - 1.65T + 11T^{2} \)
13 \( 1 + 3.36T + 13T^{2} \)
17 \( 1 + 0.467T + 17T^{2} \)
19 \( 1 - 3.22T + 19T^{2} \)
23 \( 1 - 8.94T + 23T^{2} \)
29 \( 1 + 6.26T + 29T^{2} \)
31 \( 1 + 9.23T + 31T^{2} \)
37 \( 1 - 9.23T + 37T^{2} \)
41 \( 1 + 3.41T + 41T^{2} \)
43 \( 1 - 4.41T + 43T^{2} \)
47 \( 1 - 9.35T + 47T^{2} \)
53 \( 1 - 0.573T + 53T^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 - 7.63T + 61T^{2} \)
67 \( 1 + 0.596T + 67T^{2} \)
71 \( 1 + 0.554T + 71T^{2} \)
73 \( 1 - 2.04T + 73T^{2} \)
79 \( 1 - 2.40T + 79T^{2} \)
83 \( 1 + 15.0T + 83T^{2} \)
89 \( 1 + 9.08T + 89T^{2} \)
97 \( 1 + 1.89T + 97T^{2} \)
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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

−7.25834621573529235719225440153, −7.03489576814038167285532160100, −5.91975648131135809760864526084, −5.37652170633119025062827982768, −4.52463003190870684907048557040, −3.82396398923480200610277172333, −3.12953172606560302081349864140, −2.27864963418210741362490365453, −1.12841708317702186472040900532, 0, 1.12841708317702186472040900532, 2.27864963418210741362490365453, 3.12953172606560302081349864140, 3.82396398923480200610277172333, 4.52463003190870684907048557040, 5.37652170633119025062827982768, 5.91975648131135809760864526084, 7.03489576814038167285532160100, 7.25834621573529235719225440153

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