Properties

Label 2-2e6-1.1-c23-0-7
Degree $2$
Conductor $64$
Sign $1$
Analytic cond. $214.530$
Root an. cond. $14.6468$
Motivic weight $23$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.95e5·3-s + 1.26e8·5-s − 8.35e9·7-s − 5.59e10·9-s + 2.05e11·11-s + 2.81e12·13-s − 2.47e13·15-s − 2.25e14·17-s + 1.36e14·19-s + 1.63e15·21-s − 4.06e15·23-s + 4.16e15·25-s + 2.93e16·27-s − 1.04e17·29-s − 2.09e17·31-s − 4.01e16·33-s − 1.06e18·35-s + 9.00e17·37-s − 5.50e17·39-s + 5.21e18·41-s − 5.52e18·43-s − 7.09e18·45-s − 6.66e18·47-s + 4.25e19·49-s + 4.41e19·51-s + 9.17e19·53-s + 2.60e19·55-s + ⋯
L(s)  = 1  − 0.636·3-s + 1.16·5-s − 1.59·7-s − 0.594·9-s + 0.217·11-s + 0.436·13-s − 0.739·15-s − 1.59·17-s + 0.268·19-s + 1.01·21-s − 0.889·23-s + 0.349·25-s + 1.01·27-s − 1.59·29-s − 1.47·31-s − 0.138·33-s − 1.85·35-s + 0.831·37-s − 0.277·39-s + 1.48·41-s − 0.906·43-s − 0.690·45-s − 0.392·47-s + 1.55·49-s + 1.01·51-s + 1.35·53-s + 0.252·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $1$
Analytic conductor: \(214.530\)
Root analytic conductor: \(14.6468\)
Motivic weight: \(23\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :23/2),\ 1)\)

Particular Values

\(L(12)\) \(\approx\) \(0.6277945031\)
\(L(\frac12)\) \(\approx\) \(0.6277945031\)
\(L(\frac{25}{2})\) 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 \( 1 + 1.95e5T + 9.41e10T^{2} \)
5 \( 1 - 1.26e8T + 1.19e16T^{2} \)
7 \( 1 + 8.35e9T + 2.73e19T^{2} \)
11 \( 1 - 2.05e11T + 8.95e23T^{2} \)
13 \( 1 - 2.81e12T + 4.17e25T^{2} \)
17 \( 1 + 2.25e14T + 1.99e28T^{2} \)
19 \( 1 - 1.36e14T + 2.57e29T^{2} \)
23 \( 1 + 4.06e15T + 2.08e31T^{2} \)
29 \( 1 + 1.04e17T + 4.31e33T^{2} \)
31 \( 1 + 2.09e17T + 2.00e34T^{2} \)
37 \( 1 - 9.00e17T + 1.17e36T^{2} \)
41 \( 1 - 5.21e18T + 1.24e37T^{2} \)
43 \( 1 + 5.52e18T + 3.71e37T^{2} \)
47 \( 1 + 6.66e18T + 2.87e38T^{2} \)
53 \( 1 - 9.17e19T + 4.55e39T^{2} \)
59 \( 1 - 1.00e20T + 5.36e40T^{2} \)
61 \( 1 - 3.93e19T + 1.15e41T^{2} \)
67 \( 1 + 1.31e21T + 9.99e41T^{2} \)
71 \( 1 + 2.74e21T + 3.79e42T^{2} \)
73 \( 1 + 3.27e21T + 7.18e42T^{2} \)
79 \( 1 + 6.31e21T + 4.42e43T^{2} \)
83 \( 1 + 8.53e21T + 1.37e44T^{2} \)
89 \( 1 - 1.40e22T + 6.85e44T^{2} \)
97 \( 1 + 3.66e22T + 4.96e45T^{2} \)
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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

−10.61350314611333931677447705040, −9.558064155361736697975403803207, −8.912180109078823579903023061511, −7.03809961382828953999272905912, −6.02997912164298972019180573785, −5.72699211803307057229834422246, −4.04684745605165604221837836278, −2.82231169441488429126371525455, −1.81074299104509396611702191848, −0.32086632407995379261035611959, 0.32086632407995379261035611959, 1.81074299104509396611702191848, 2.82231169441488429126371525455, 4.04684745605165604221837836278, 5.72699211803307057229834422246, 6.02997912164298972019180573785, 7.03809961382828953999272905912, 8.912180109078823579903023061511, 9.558064155361736697975403803207, 10.61350314611333931677447705040

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