Properties

Label 2-2e6-1.1-c3-0-3
Degree $2$
Conductor $64$
Sign $-1$
Analytic cond. $3.77612$
Root an. cond. $1.94322$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s + 2·5-s − 24·7-s − 11·9-s − 44·11-s − 22·13-s − 8·15-s + 50·17-s + 44·19-s + 96·21-s + 56·23-s − 121·25-s + 152·27-s − 198·29-s + 160·31-s + 176·33-s − 48·35-s + 162·37-s + 88·39-s − 198·41-s + 52·43-s − 22·45-s − 528·47-s + 233·49-s − 200·51-s + 242·53-s − 88·55-s + ⋯
L(s)  = 1  − 0.769·3-s + 0.178·5-s − 1.29·7-s − 0.407·9-s − 1.20·11-s − 0.469·13-s − 0.137·15-s + 0.713·17-s + 0.531·19-s + 0.997·21-s + 0.507·23-s − 0.967·25-s + 1.08·27-s − 1.26·29-s + 0.926·31-s + 0.928·33-s − 0.231·35-s + 0.719·37-s + 0.361·39-s − 0.754·41-s + 0.184·43-s − 0.0728·45-s − 1.63·47-s + 0.679·49-s − 0.549·51-s + 0.627·53-s − 0.215·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $-1$
Analytic conductor: \(3.77612\)
Root analytic conductor: \(1.94322\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 64,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 4 T + p^{3} T^{2} \)
5 \( 1 - 2 T + p^{3} T^{2} \)
7 \( 1 + 24 T + p^{3} T^{2} \)
11 \( 1 + 4 p T + p^{3} T^{2} \)
13 \( 1 + 22 T + p^{3} T^{2} \)
17 \( 1 - 50 T + p^{3} T^{2} \)
19 \( 1 - 44 T + p^{3} T^{2} \)
23 \( 1 - 56 T + p^{3} T^{2} \)
29 \( 1 + 198 T + p^{3} T^{2} \)
31 \( 1 - 160 T + p^{3} T^{2} \)
37 \( 1 - 162 T + p^{3} T^{2} \)
41 \( 1 + 198 T + p^{3} T^{2} \)
43 \( 1 - 52 T + p^{3} T^{2} \)
47 \( 1 + 528 T + p^{3} T^{2} \)
53 \( 1 - 242 T + p^{3} T^{2} \)
59 \( 1 + 668 T + p^{3} T^{2} \)
61 \( 1 + 550 T + p^{3} T^{2} \)
67 \( 1 - 188 T + p^{3} T^{2} \)
71 \( 1 + 728 T + p^{3} T^{2} \)
73 \( 1 - 154 T + p^{3} T^{2} \)
79 \( 1 - 656 T + p^{3} T^{2} \)
83 \( 1 - 236 T + p^{3} T^{2} \)
89 \( 1 - 714 T + p^{3} T^{2} \)
97 \( 1 + 478 T + p^{3} T^{2} \)
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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

−13.64990555602677464106126417049, −12.73092037334301660970883830398, −11.67336650869207088746891442217, −10.39838762153357305727761130279, −9.483388672462055925742877333572, −7.73849281225555610878058287980, −6.26596002171701478217565349335, −5.24179280659455016334003035556, −3.03322343954749351036954026424, 0, 3.03322343954749351036954026424, 5.24179280659455016334003035556, 6.26596002171701478217565349335, 7.73849281225555610878058287980, 9.483388672462055925742877333572, 10.39838762153357305727761130279, 11.67336650869207088746891442217, 12.73092037334301660970883830398, 13.64990555602677464106126417049

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