Properties

Label 2-24-1.1-c21-0-1
Degree $2$
Conductor $24$
Sign $1$
Analytic cond. $67.0745$
Root an. cond. $8.18990$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.90e4·3-s − 3.51e7·5-s + 2.43e8·7-s + 3.48e9·9-s − 5.47e9·11-s − 2.75e11·13-s − 2.07e12·15-s − 7.88e12·17-s − 1.71e13·19-s + 1.43e13·21-s + 1.02e14·23-s + 7.61e14·25-s + 2.05e14·27-s + 3.25e15·29-s − 5.25e15·31-s − 3.23e14·33-s − 8.56e15·35-s − 1.32e15·37-s − 1.62e16·39-s + 1.25e17·41-s − 8.18e16·43-s − 1.22e17·45-s + 3.29e17·47-s − 4.99e17·49-s − 4.65e17·51-s + 1.26e18·53-s + 1.92e17·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.61·5-s + 0.325·7-s + 0.333·9-s − 0.0635·11-s − 0.554·13-s − 0.930·15-s − 0.948·17-s − 0.642·19-s + 0.187·21-s + 0.514·23-s + 1.59·25-s + 0.192·27-s + 1.43·29-s − 1.15·31-s − 0.0367·33-s − 0.524·35-s − 0.0454·37-s − 0.320·39-s + 1.46·41-s − 0.577·43-s − 0.537·45-s + 0.913·47-s − 0.893·49-s − 0.547·51-s + 0.990·53-s + 0.102·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24\)    =    \(2^{3} \cdot 3\)
Sign: $1$
Analytic conductor: \(67.0745\)
Root analytic conductor: \(8.18990\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 24,\ (\ :21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(1.542269376\)
\(L(\frac12)\) \(\approx\) \(1.542269376\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 5.90e4T \)
good5 \( 1 + 3.51e7T + 4.76e14T^{2} \)
7 \( 1 - 2.43e8T + 5.58e17T^{2} \)
11 \( 1 + 5.47e9T + 7.40e21T^{2} \)
13 \( 1 + 2.75e11T + 2.47e23T^{2} \)
17 \( 1 + 7.88e12T + 6.90e25T^{2} \)
19 \( 1 + 1.71e13T + 7.14e26T^{2} \)
23 \( 1 - 1.02e14T + 3.94e28T^{2} \)
29 \( 1 - 3.25e15T + 5.13e30T^{2} \)
31 \( 1 + 5.25e15T + 2.08e31T^{2} \)
37 \( 1 + 1.32e15T + 8.55e32T^{2} \)
41 \( 1 - 1.25e17T + 7.38e33T^{2} \)
43 \( 1 + 8.18e16T + 2.00e34T^{2} \)
47 \( 1 - 3.29e17T + 1.30e35T^{2} \)
53 \( 1 - 1.26e18T + 1.62e36T^{2} \)
59 \( 1 - 5.01e18T + 1.54e37T^{2} \)
61 \( 1 - 7.90e18T + 3.10e37T^{2} \)
67 \( 1 + 9.13e18T + 2.22e38T^{2} \)
71 \( 1 + 3.27e19T + 7.52e38T^{2} \)
73 \( 1 - 2.66e19T + 1.34e39T^{2} \)
79 \( 1 - 1.51e20T + 7.08e39T^{2} \)
83 \( 1 - 2.78e20T + 1.99e40T^{2} \)
89 \( 1 + 1.69e20T + 8.65e40T^{2} \)
97 \( 1 - 5.36e20T + 5.27e41T^{2} \)
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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

−12.94066493649598194601934906117, −11.80347117139358426483154909311, −10.69613369679589813169423232568, −8.919380742769244515081597249875, −7.952461587429409395727375844113, −6.92402663277837857128896666014, −4.73914838199475970225262588449, −3.77067262604501734440173626088, −2.40436483292797185410503477636, −0.62248433116497439205336338380, 0.62248433116497439205336338380, 2.40436483292797185410503477636, 3.77067262604501734440173626088, 4.73914838199475970225262588449, 6.92402663277837857128896666014, 7.952461587429409395727375844113, 8.919380742769244515081597249875, 10.69613369679589813169423232568, 11.80347117139358426483154909311, 12.94066493649598194601934906117

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