Properties

Label 2-2e6-1.1-c7-0-3
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 $0$

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: \(0\)
Selberg data: \((2,\ 64,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(2.167859558\)
\(L(\frac12)\) \(\approx\) \(2.167859558\)
\(L(\frac{9}{2})\) 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

−13.75937780491947542560160016600, −12.15957133411481030775995390275, −11.26247528828948581507771474148, −10.09551263100127005692407854544, −8.774333268076106561899272521381, −7.55407852645638831676267158334, −5.85363689269287980367864054708, −4.94217347209523946874423427174, −2.78147749519244964953202491872, −1.11402698778958475176091828630, 1.11402698778958475176091828630, 2.78147749519244964953202491872, 4.94217347209523946874423427174, 5.85363689269287980367864054708, 7.55407852645638831676267158334, 8.774333268076106561899272521381, 10.09551263100127005692407854544, 11.26247528828948581507771474148, 12.15957133411481030775995390275, 13.75937780491947542560160016600

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