Properties

Label 2-2e6-1.1-c23-0-34
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.88e5·3-s − 1.05e8·5-s − 3.81e9·7-s + 5.67e10·9-s + 2.52e11·11-s + 3.59e12·13-s − 4.08e13·15-s + 2.34e14·17-s − 6.23e14·19-s − 1.48e15·21-s + 3.58e15·23-s − 8.82e14·25-s − 1.45e16·27-s + 2.05e16·29-s − 1.36e17·31-s + 9.79e16·33-s + 4.00e17·35-s + 1.23e18·37-s + 1.39e18·39-s + 1.40e18·41-s + 2.18e17·43-s − 5.96e18·45-s + 8.67e18·47-s − 1.28e19·49-s + 9.09e19·51-s + 7.63e19·53-s − 2.64e19·55-s + ⋯
L(s)  = 1  + 1.26·3-s − 0.962·5-s − 0.728·7-s + 0.602·9-s + 0.266·11-s + 0.555·13-s − 1.21·15-s + 1.65·17-s − 1.22·19-s − 0.922·21-s + 0.785·23-s − 0.0740·25-s − 0.502·27-s + 0.313·29-s − 0.963·31-s + 0.337·33-s + 0.701·35-s + 1.14·37-s + 0.703·39-s + 0.397·41-s + 0.0359·43-s − 0.580·45-s + 0.512·47-s − 0.469·49-s + 2.09·51-s + 1.13·53-s − 0.256·55-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: $\chi_{64} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 64,\ (\ :23/2),\ -1)\)

Particular Values

\(L(12)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{25}{2})\) 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 \( 1 - 3.88e5T + 9.41e10T^{2} \)
5 \( 1 + 1.05e8T + 1.19e16T^{2} \)
7 \( 1 + 3.81e9T + 2.73e19T^{2} \)
11 \( 1 - 2.52e11T + 8.95e23T^{2} \)
13 \( 1 - 3.59e12T + 4.17e25T^{2} \)
17 \( 1 - 2.34e14T + 1.99e28T^{2} \)
19 \( 1 + 6.23e14T + 2.57e29T^{2} \)
23 \( 1 - 3.58e15T + 2.08e31T^{2} \)
29 \( 1 - 2.05e16T + 4.31e33T^{2} \)
31 \( 1 + 1.36e17T + 2.00e34T^{2} \)
37 \( 1 - 1.23e18T + 1.17e36T^{2} \)
41 \( 1 - 1.40e18T + 1.24e37T^{2} \)
43 \( 1 - 2.18e17T + 3.71e37T^{2} \)
47 \( 1 - 8.67e18T + 2.87e38T^{2} \)
53 \( 1 - 7.63e19T + 4.55e39T^{2} \)
59 \( 1 + 1.01e18T + 5.36e40T^{2} \)
61 \( 1 + 2.87e20T + 1.15e41T^{2} \)
67 \( 1 - 1.47e21T + 9.99e41T^{2} \)
71 \( 1 + 7.64e20T + 3.79e42T^{2} \)
73 \( 1 + 3.49e21T + 7.18e42T^{2} \)
79 \( 1 + 1.02e22T + 4.42e43T^{2} \)
83 \( 1 + 7.71e21T + 1.37e44T^{2} \)
89 \( 1 - 4.58e21T + 6.85e44T^{2} \)
97 \( 1 + 1.13e23T + 4.96e45T^{2} \)
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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

−9.895293886851635573468593754445, −8.891119794484000977659631706985, −8.078431405365765655048705985109, −7.20076247862321241337333359340, −5.85667632587706229574287450487, −4.13672942642286505432777441969, −3.46953405887859205323107349635, −2.63736641430245155695974597026, −1.24201727822472133475307307980, 0, 1.24201727822472133475307307980, 2.63736641430245155695974597026, 3.46953405887859205323107349635, 4.13672942642286505432777441969, 5.85667632587706229574287450487, 7.20076247862321241337333359340, 8.078431405365765655048705985109, 8.891119794484000977659631706985, 9.895293886851635573468593754445

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