Properties

Label 2-2e6-1.1-c23-0-12
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  − 1.56e5·3-s − 1.90e8·5-s − 4.80e9·7-s − 6.97e10·9-s − 1.33e12·11-s − 9.05e12·13-s + 2.97e13·15-s + 4.05e13·17-s + 5.13e14·19-s + 7.50e14·21-s + 3.70e15·23-s + 2.44e16·25-s + 2.56e16·27-s − 5.34e15·29-s − 1.61e17·31-s + 2.08e17·33-s + 9.16e17·35-s − 9.84e17·37-s + 1.41e18·39-s + 3.74e18·41-s + 7.77e17·43-s + 1.32e19·45-s + 2.36e19·47-s − 4.26e18·49-s − 6.32e18·51-s − 5.09e19·53-s + 2.54e20·55-s + ⋯
L(s)  = 1  − 0.509·3-s − 1.74·5-s − 0.918·7-s − 0.740·9-s − 1.40·11-s − 1.40·13-s + 0.888·15-s + 0.286·17-s + 1.01·19-s + 0.467·21-s + 0.810·23-s + 2.04·25-s + 0.886·27-s − 0.0812·29-s − 1.14·31-s + 0.716·33-s + 1.60·35-s − 0.909·37-s + 0.713·39-s + 1.06·41-s + 0.127·43-s + 1.29·45-s + 1.39·47-s − 0.155·49-s − 0.145·51-s − 0.755·53-s + 2.45·55-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: \(1\)
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 \( 1 + 1.56e5T + 9.41e10T^{2} \)
5 \( 1 + 1.90e8T + 1.19e16T^{2} \)
7 \( 1 + 4.80e9T + 2.73e19T^{2} \)
11 \( 1 + 1.33e12T + 8.95e23T^{2} \)
13 \( 1 + 9.05e12T + 4.17e25T^{2} \)
17 \( 1 - 4.05e13T + 1.99e28T^{2} \)
19 \( 1 - 5.13e14T + 2.57e29T^{2} \)
23 \( 1 - 3.70e15T + 2.08e31T^{2} \)
29 \( 1 + 5.34e15T + 4.31e33T^{2} \)
31 \( 1 + 1.61e17T + 2.00e34T^{2} \)
37 \( 1 + 9.84e17T + 1.17e36T^{2} \)
41 \( 1 - 3.74e18T + 1.24e37T^{2} \)
43 \( 1 - 7.77e17T + 3.71e37T^{2} \)
47 \( 1 - 2.36e19T + 2.87e38T^{2} \)
53 \( 1 + 5.09e19T + 4.55e39T^{2} \)
59 \( 1 - 3.99e20T + 5.36e40T^{2} \)
61 \( 1 + 4.81e20T + 1.15e41T^{2} \)
67 \( 1 + 1.52e21T + 9.99e41T^{2} \)
71 \( 1 + 1.07e21T + 3.79e42T^{2} \)
73 \( 1 - 1.32e21T + 7.18e42T^{2} \)
79 \( 1 - 1.09e21T + 4.42e43T^{2} \)
83 \( 1 + 1.68e22T + 1.37e44T^{2} \)
89 \( 1 + 1.71e22T + 6.85e44T^{2} \)
97 \( 1 - 7.38e22T + 4.96e45T^{2} \)
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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

−10.34012632091204019969653495423, −8.989920236553066610931570248413, −7.68207874934686475857120383996, −7.19755416615285294748383182697, −5.59028978554926910398488640525, −4.73714799434028549015697012888, −3.35442163735016874068001227676, −2.72470381915829223371576169257, −0.57799906427386718623503042335, 0, 0.57799906427386718623503042335, 2.72470381915829223371576169257, 3.35442163735016874068001227676, 4.73714799434028549015697012888, 5.59028978554926910398488640525, 7.19755416615285294748383182697, 7.68207874934686475857120383996, 8.989920236553066610931570248413, 10.34012632091204019969653495423

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