Properties

Label 2-2e6-1.1-c23-0-21
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  + 2.53e5·3-s + 1.36e8·5-s − 4.69e9·7-s − 3.00e10·9-s + 2.44e11·11-s + 9.00e12·13-s + 3.46e13·15-s + 1.66e14·17-s + 6.92e14·19-s − 1.18e15·21-s + 8.91e13·23-s + 6.81e15·25-s − 3.14e16·27-s − 7.47e16·29-s + 1.11e17·31-s + 6.19e16·33-s − 6.42e17·35-s − 1.38e18·37-s + 2.28e18·39-s − 5.25e17·41-s + 1.20e19·43-s − 4.11e18·45-s − 8.11e18·47-s − 5.34e18·49-s + 4.21e19·51-s + 9.12e19·53-s + 3.34e19·55-s + ⋯
L(s)  = 1  + 0.824·3-s + 1.25·5-s − 0.897·7-s − 0.319·9-s + 0.258·11-s + 1.39·13-s + 1.03·15-s + 1.17·17-s + 1.36·19-s − 0.739·21-s + 0.0195·23-s + 0.571·25-s − 1.08·27-s − 1.13·29-s + 0.787·31-s + 0.213·33-s − 1.12·35-s − 1.28·37-s + 1.14·39-s − 0.149·41-s + 1.98·43-s − 0.400·45-s − 0.478·47-s − 0.195·49-s + 0.971·51-s + 1.35·53-s + 0.324·55-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

\(L(12)\) \(\approx\) \(4.378600845\)
\(L(\frac12)\) \(\approx\) \(4.378600845\)
\(L(\frac{25}{2})\) 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 \( 1 - 2.53e5T + 9.41e10T^{2} \)
5 \( 1 - 1.36e8T + 1.19e16T^{2} \)
7 \( 1 + 4.69e9T + 2.73e19T^{2} \)
11 \( 1 - 2.44e11T + 8.95e23T^{2} \)
13 \( 1 - 9.00e12T + 4.17e25T^{2} \)
17 \( 1 - 1.66e14T + 1.99e28T^{2} \)
19 \( 1 - 6.92e14T + 2.57e29T^{2} \)
23 \( 1 - 8.91e13T + 2.08e31T^{2} \)
29 \( 1 + 7.47e16T + 4.31e33T^{2} \)
31 \( 1 - 1.11e17T + 2.00e34T^{2} \)
37 \( 1 + 1.38e18T + 1.17e36T^{2} \)
41 \( 1 + 5.25e17T + 1.24e37T^{2} \)
43 \( 1 - 1.20e19T + 3.71e37T^{2} \)
47 \( 1 + 8.11e18T + 2.87e38T^{2} \)
53 \( 1 - 9.12e19T + 4.55e39T^{2} \)
59 \( 1 - 4.14e20T + 5.36e40T^{2} \)
61 \( 1 + 6.24e20T + 1.15e41T^{2} \)
67 \( 1 + 3.72e20T + 9.99e41T^{2} \)
71 \( 1 + 2.77e21T + 3.79e42T^{2} \)
73 \( 1 - 1.39e21T + 7.18e42T^{2} \)
79 \( 1 + 4.37e21T + 4.42e43T^{2} \)
83 \( 1 - 1.10e21T + 1.37e44T^{2} \)
89 \( 1 - 5.05e22T + 6.85e44T^{2} \)
97 \( 1 + 7.16e21T + 4.96e45T^{2} \)
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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

−10.32780490430107163776848499849, −9.465109178788060710855980919499, −8.799024221528586227064290377064, −7.48906071529783151281292292141, −6.11075874511094559906377326811, −5.55156382274342296091545785224, −3.62886259382546418988883993901, −3.01480932833622032853319404565, −1.84248036700612153862678437219, −0.855969042453938478598309234132, 0.855969042453938478598309234132, 1.84248036700612153862678437219, 3.01480932833622032853319404565, 3.62886259382546418988883993901, 5.55156382274342296091545785224, 6.11075874511094559906377326811, 7.48906071529783151281292292141, 8.799024221528586227064290377064, 9.465109178788060710855980919499, 10.32780490430107163776848499849

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