Properties

Label 2-9072-1.1-c1-0-70
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  − 3.74·5-s − 7-s + 3.65·11-s − 5.54·13-s − 7.20·17-s + 3.30·19-s + 4.98·23-s + 9.05·25-s + 0.490·29-s + 3.89·31-s + 3.74·35-s + 7.89·37-s + 4.76·41-s + 1.60·43-s − 9.63·47-s + 49-s − 8.03·53-s − 13.6·55-s + 1.50·59-s − 2.08·61-s + 20.7·65-s − 3.40·67-s + 10.7·71-s + 9.83·73-s − 3.65·77-s − 3.72·79-s + 11.3·83-s + ⋯
L(s)  = 1  − 1.67·5-s − 0.377·7-s + 1.10·11-s − 1.53·13-s − 1.74·17-s + 0.758·19-s + 1.04·23-s + 1.81·25-s + 0.0911·29-s + 0.699·31-s + 0.633·35-s + 1.29·37-s + 0.744·41-s + 0.244·43-s − 1.40·47-s + 0.142·49-s − 1.10·53-s − 1.84·55-s + 0.196·59-s − 0.267·61-s + 2.57·65-s − 0.415·67-s + 1.27·71-s + 1.15·73-s − 0.416·77-s − 0.419·79-s + 1.24·83-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 \( 1 + 3.74T + 5T^{2} \)
11 \( 1 - 3.65T + 11T^{2} \)
13 \( 1 + 5.54T + 13T^{2} \)
17 \( 1 + 7.20T + 17T^{2} \)
19 \( 1 - 3.30T + 19T^{2} \)
23 \( 1 - 4.98T + 23T^{2} \)
29 \( 1 - 0.490T + 29T^{2} \)
31 \( 1 - 3.89T + 31T^{2} \)
37 \( 1 - 7.89T + 37T^{2} \)
41 \( 1 - 4.76T + 41T^{2} \)
43 \( 1 - 1.60T + 43T^{2} \)
47 \( 1 + 9.63T + 47T^{2} \)
53 \( 1 + 8.03T + 53T^{2} \)
59 \( 1 - 1.50T + 59T^{2} \)
61 \( 1 + 2.08T + 61T^{2} \)
67 \( 1 + 3.40T + 67T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 - 9.83T + 73T^{2} \)
79 \( 1 + 3.72T + 79T^{2} \)
83 \( 1 - 11.3T + 83T^{2} \)
89 \( 1 - 7.14T + 89T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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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

−7.39377762765123840362827921090, −6.78994158949195607262944988492, −6.32759817256378850141849280004, −4.95149101726470578627842904835, −4.59664297765263944034230612301, −3.90729873446771743250220512164, −3.13057923133597145691614270946, −2.38413483723863661956246056541, −0.958030353699659687071041484775, 0, 0.958030353699659687071041484775, 2.38413483723863661956246056541, 3.13057923133597145691614270946, 3.90729873446771743250220512164, 4.59664297765263944034230612301, 4.95149101726470578627842904835, 6.32759817256378850141849280004, 6.78994158949195607262944988492, 7.39377762765123840362827921090

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