Properties

Label 2-9072-1.1-c1-0-50
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.44·5-s + 7-s + 2·11-s + 4.89·13-s − 2·17-s − 2.55·19-s − 23-s − 2.89·25-s + 6.89·29-s − 6·31-s + 1.44·35-s + 11.7·37-s − 9.79·41-s − 6.89·43-s + 9.79·47-s + 49-s + 10.8·53-s + 2.89·55-s − 2·59-s + 6.55·61-s + 7.10·65-s + 12.8·67-s + 0.101·71-s − 6.89·73-s + 2·77-s + 1.89·79-s + 2·83-s + ⋯
L(s)  = 1  + 0.648·5-s + 0.377·7-s + 0.603·11-s + 1.35·13-s − 0.485·17-s − 0.585·19-s − 0.208·23-s − 0.579·25-s + 1.28·29-s − 1.07·31-s + 0.245·35-s + 1.93·37-s − 1.53·41-s − 1.05·43-s + 1.42·47-s + 0.142·49-s + 1.49·53-s + 0.390·55-s − 0.260·59-s + 0.838·61-s + 0.880·65-s + 1.57·67-s + 0.0119·71-s − 0.807·73-s + 0.227·77-s + 0.213·79-s + 0.219·83-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: $\chi_{9072} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9072,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.827274040\)
\(L(\frac12)\) \(\approx\) \(2.827274040\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.44T + 5T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 4.89T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 2.55T + 19T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 - 6.89T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 - 11.7T + 37T^{2} \)
41 \( 1 + 9.79T + 41T^{2} \)
43 \( 1 + 6.89T + 43T^{2} \)
47 \( 1 - 9.79T + 47T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 + 2T + 59T^{2} \)
61 \( 1 - 6.55T + 61T^{2} \)
67 \( 1 - 12.8T + 67T^{2} \)
71 \( 1 - 0.101T + 71T^{2} \)
73 \( 1 + 6.89T + 73T^{2} \)
79 \( 1 - 1.89T + 79T^{2} \)
83 \( 1 - 2T + 83T^{2} \)
89 \( 1 - 16.8T + 89T^{2} \)
97 \( 1 - 2.89T + 97T^{2} \)
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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

−7.82038464544903319263206966495, −6.87081939073847648021393907801, −6.33257667762718718182413818760, −5.80495920410502349463955589595, −5.00412268718490120571879899049, −4.12736257027854662369664200828, −3.61614667015159819703147009218, −2.45550484320101668634316809966, −1.75874268382395736534777293173, −0.847117957674979075798855078951, 0.847117957674979075798855078951, 1.75874268382395736534777293173, 2.45550484320101668634316809966, 3.61614667015159819703147009218, 4.12736257027854662369664200828, 5.00412268718490120571879899049, 5.80495920410502349463955589595, 6.33257667762718718182413818760, 6.87081939073847648021393907801, 7.82038464544903319263206966495

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