Properties

Label 2-2e6-1.1-c5-0-5
Degree $2$
Conductor $64$
Sign $-1$
Analytic cond. $10.2645$
Root an. cond. $3.20383$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 20·3-s + 74·5-s − 24·7-s + 157·9-s − 124·11-s − 478·13-s − 1.48e3·15-s − 1.19e3·17-s − 3.04e3·19-s + 480·21-s + 184·23-s + 2.35e3·25-s + 1.72e3·27-s + 3.28e3·29-s − 5.72e3·31-s + 2.48e3·33-s − 1.77e3·35-s − 1.03e4·37-s + 9.56e3·39-s − 8.88e3·41-s + 9.18e3·43-s + 1.16e4·45-s + 2.36e4·47-s − 1.62e4·49-s + 2.39e4·51-s − 1.16e4·53-s − 9.17e3·55-s + ⋯
L(s)  = 1  − 1.28·3-s + 1.32·5-s − 0.185·7-s + 0.646·9-s − 0.308·11-s − 0.784·13-s − 1.69·15-s − 1.00·17-s − 1.93·19-s + 0.237·21-s + 0.0725·23-s + 0.752·25-s + 0.454·27-s + 0.724·29-s − 1.07·31-s + 0.396·33-s − 0.245·35-s − 1.24·37-s + 1.00·39-s − 0.825·41-s + 0.757·43-s + 0.855·45-s + 1.56·47-s − 0.965·49-s + 1.28·51-s − 0.571·53-s − 0.409·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $-1$
Analytic conductor: \(10.2645\)
Root analytic conductor: \(3.20383\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 64,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 + 20 T + p^{5} T^{2} \)
5 \( 1 - 74 T + p^{5} T^{2} \)
7 \( 1 + 24 T + p^{5} T^{2} \)
11 \( 1 + 124 T + p^{5} T^{2} \)
13 \( 1 + 478 T + p^{5} T^{2} \)
17 \( 1 + 1198 T + p^{5} T^{2} \)
19 \( 1 + 3044 T + p^{5} T^{2} \)
23 \( 1 - 8 p T + p^{5} T^{2} \)
29 \( 1 - 3282 T + p^{5} T^{2} \)
31 \( 1 + 5728 T + p^{5} T^{2} \)
37 \( 1 + 10326 T + p^{5} T^{2} \)
41 \( 1 + 8886 T + p^{5} T^{2} \)
43 \( 1 - 9188 T + p^{5} T^{2} \)
47 \( 1 - 23664 T + p^{5} T^{2} \)
53 \( 1 + 11686 T + p^{5} T^{2} \)
59 \( 1 + 16876 T + p^{5} T^{2} \)
61 \( 1 - 18482 T + p^{5} T^{2} \)
67 \( 1 - 15532 T + p^{5} T^{2} \)
71 \( 1 + 31960 T + p^{5} T^{2} \)
73 \( 1 + 4886 T + p^{5} T^{2} \)
79 \( 1 - 44560 T + p^{5} T^{2} \)
83 \( 1 + 67364 T + p^{5} T^{2} \)
89 \( 1 - 71994 T + p^{5} T^{2} \)
97 \( 1 - 48866 T + p^{5} T^{2} \)
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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

−13.15991808413740951626359527000, −12.33617781348477694030161382498, −10.91486176652043111891051453818, −10.21667213252376832058300620725, −8.878750413009208580531506918918, −6.79649587461587753127146464477, −5.91695460441500439551887933781, −4.78614279604806690123249976975, −2.13791195355192551764797556587, 0, 2.13791195355192551764797556587, 4.78614279604806690123249976975, 5.91695460441500439551887933781, 6.79649587461587753127146464477, 8.878750413009208580531506918918, 10.21667213252376832058300620725, 10.91486176652043111891051453818, 12.33617781348477694030161382498, 13.15991808413740951626359527000

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