Properties

Label 2-2e6-1.1-c7-0-10
Degree $2$
Conductor $64$
Sign $-1$
Analytic cond. $19.9926$
Root an. cond. $4.47131$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 12·3-s + 210·5-s − 1.01e3·7-s − 2.04e3·9-s + 1.09e3·11-s − 1.38e3·13-s + 2.52e3·15-s + 1.47e4·17-s − 3.99e4·19-s − 1.21e4·21-s − 6.87e4·23-s − 3.40e4·25-s − 5.07e4·27-s + 1.02e5·29-s − 2.27e5·31-s + 1.31e4·33-s − 2.13e5·35-s − 1.60e5·37-s − 1.65e4·39-s + 1.08e4·41-s − 6.30e5·43-s − 4.29e5·45-s − 4.72e5·47-s + 2.08e5·49-s + 1.76e5·51-s + 1.49e6·53-s + 2.29e5·55-s + ⋯
L(s)  = 1  + 0.256·3-s + 0.751·5-s − 1.11·7-s − 0.934·9-s + 0.247·11-s − 0.174·13-s + 0.192·15-s + 0.725·17-s − 1.33·19-s − 0.287·21-s − 1.17·23-s − 0.435·25-s − 0.496·27-s + 0.780·29-s − 1.37·31-s + 0.0634·33-s − 0.841·35-s − 0.521·37-s − 0.0447·39-s + 0.0245·41-s − 1.20·43-s − 0.701·45-s − 0.664·47-s + 0.253·49-s + 0.186·51-s + 1.37·53-s + 0.185·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $-1$
Analytic conductor: \(19.9926\)
Root analytic conductor: \(4.47131\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 64,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 - 4 p T + p^{7} T^{2} \)
5 \( 1 - 42 p T + p^{7} T^{2} \)
7 \( 1 + 1016 T + p^{7} T^{2} \)
11 \( 1 - 1092 T + p^{7} T^{2} \)
13 \( 1 + 1382 T + p^{7} T^{2} \)
17 \( 1 - 14706 T + p^{7} T^{2} \)
19 \( 1 + 39940 T + p^{7} T^{2} \)
23 \( 1 + 68712 T + p^{7} T^{2} \)
29 \( 1 - 102570 T + p^{7} T^{2} \)
31 \( 1 + 227552 T + p^{7} T^{2} \)
37 \( 1 + 160526 T + p^{7} T^{2} \)
41 \( 1 - 10842 T + p^{7} T^{2} \)
43 \( 1 + 630748 T + p^{7} T^{2} \)
47 \( 1 + 472656 T + p^{7} T^{2} \)
53 \( 1 - 1494018 T + p^{7} T^{2} \)
59 \( 1 - 2640660 T + p^{7} T^{2} \)
61 \( 1 + 827702 T + p^{7} T^{2} \)
67 \( 1 + 126004 T + p^{7} T^{2} \)
71 \( 1 - 1414728 T + p^{7} T^{2} \)
73 \( 1 - 980282 T + p^{7} T^{2} \)
79 \( 1 - 3566800 T + p^{7} T^{2} \)
83 \( 1 - 5672892 T + p^{7} T^{2} \)
89 \( 1 + 11951190 T + p^{7} T^{2} \)
97 \( 1 - 8682146 T + p^{7} T^{2} \)
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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

−13.01846102809393928726060638743, −11.91122970355665011467074566381, −10.37040586628085179362757926980, −9.474636582515927967495537204868, −8.318314244162567645865277423011, −6.59794190861557605924045624245, −5.62465193212774232740496420102, −3.59395948552495553199392527352, −2.17649061350948825402264830948, 0, 2.17649061350948825402264830948, 3.59395948552495553199392527352, 5.62465193212774232740496420102, 6.59794190861557605924045624245, 8.318314244162567645865277423011, 9.474636582515927967495537204868, 10.37040586628085179362757926980, 11.91122970355665011467074566381, 13.01846102809393928726060638743

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