Properties

Degree $2$
Conductor $64$
Sign $-1$
Motivic weight $19$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.05e4·3-s − 2.22e6·5-s − 1.60e8·7-s + 1.39e9·9-s − 1.17e10·11-s + 2.44e9·13-s + 1.12e11·15-s + 5.98e11·17-s − 2.51e12·19-s + 8.13e12·21-s + 6.35e11·23-s − 1.41e13·25-s − 1.16e13·27-s + 4.78e13·29-s + 2.03e14·31-s + 5.94e14·33-s + 3.57e14·35-s − 8.95e14·37-s − 1.23e14·39-s − 4.61e14·41-s − 2.21e15·43-s − 3.09e15·45-s + 1.04e16·47-s + 1.45e16·49-s − 3.02e16·51-s + 3.43e16·53-s + 2.61e16·55-s + ⋯
L(s)  = 1  − 1.48·3-s − 0.509·5-s − 1.50·7-s + 1.19·9-s − 1.50·11-s + 0.0638·13-s + 0.754·15-s + 1.22·17-s − 1.78·19-s + 2.23·21-s + 0.0735·23-s − 0.740·25-s − 0.294·27-s + 0.611·29-s + 1.37·31-s + 2.22·33-s + 0.767·35-s − 1.13·37-s − 0.0946·39-s − 0.219·41-s − 0.672·43-s − 0.610·45-s + 1.36·47-s + 1.27·49-s − 1.81·51-s + 1.43·53-s + 0.765·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $-1$
Motivic weight: \(19\)
Character: $\chi_{64} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 64,\ (\ :19/2),\ -1)\)

Particular Values

\(L(10)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{21}{2})\) 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 \( 1 + 5.05e4T + 1.16e9T^{2} \)
5 \( 1 + 2.22e6T + 1.90e13T^{2} \)
7 \( 1 + 1.60e8T + 1.13e16T^{2} \)
11 \( 1 + 1.17e10T + 6.11e19T^{2} \)
13 \( 1 - 2.44e9T + 1.46e21T^{2} \)
17 \( 1 - 5.98e11T + 2.39e23T^{2} \)
19 \( 1 + 2.51e12T + 1.97e24T^{2} \)
23 \( 1 - 6.35e11T + 7.46e25T^{2} \)
29 \( 1 - 4.78e13T + 6.10e27T^{2} \)
31 \( 1 - 2.03e14T + 2.16e28T^{2} \)
37 \( 1 + 8.95e14T + 6.24e29T^{2} \)
41 \( 1 + 4.61e14T + 4.39e30T^{2} \)
43 \( 1 + 2.21e15T + 1.08e31T^{2} \)
47 \( 1 - 1.04e16T + 5.88e31T^{2} \)
53 \( 1 - 3.43e16T + 5.77e32T^{2} \)
59 \( 1 - 9.13e16T + 4.42e33T^{2} \)
61 \( 1 - 1.19e17T + 8.34e33T^{2} \)
67 \( 1 + 1.98e17T + 4.95e34T^{2} \)
71 \( 1 - 1.38e17T + 1.49e35T^{2} \)
73 \( 1 + 3.63e16T + 2.53e35T^{2} \)
79 \( 1 - 4.36e17T + 1.13e36T^{2} \)
83 \( 1 - 4.52e17T + 2.90e36T^{2} \)
89 \( 1 - 3.30e18T + 1.09e37T^{2} \)
97 \( 1 + 6.03e18T + 5.60e37T^{2} \)
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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

−10.47455153486627674365912182269, −10.13656861929590377405473625328, −8.326665999890917032298499845086, −6.98297267199730590994431126005, −6.08511826082408160116640113988, −5.19964944543956427413143200528, −3.87957257274799068608341918374, −2.58899255239411414933578793451, −0.67929799869599848299107229818, 0, 0.67929799869599848299107229818, 2.58899255239411414933578793451, 3.87957257274799068608341918374, 5.19964944543956427413143200528, 6.08511826082408160116640113988, 6.98297267199730590994431126005, 8.326665999890917032298499845086, 10.13656861929590377405473625328, 10.47455153486627674365912182269

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