Properties

Label 2-24-1.1-c21-0-6
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.90e4·3-s − 8.90e6·5-s + 5.87e8·7-s + 3.48e9·9-s − 1.42e11·11-s + 1.49e11·13-s + 5.26e11·15-s + 9.95e12·17-s − 5.02e12·19-s − 3.47e13·21-s + 3.25e14·23-s − 3.97e14·25-s − 2.05e14·27-s + 1.07e15·29-s + 3.98e15·31-s + 8.40e15·33-s − 5.23e15·35-s − 3.35e16·37-s − 8.80e15·39-s + 4.54e16·41-s − 8.38e16·43-s − 3.10e16·45-s − 5.34e16·47-s − 2.13e17·49-s − 5.87e17·51-s + 1.14e18·53-s + 1.26e18·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.407·5-s + 0.786·7-s + 0.333·9-s − 1.65·11-s + 0.299·13-s + 0.235·15-s + 1.19·17-s − 0.188·19-s − 0.454·21-s + 1.63·23-s − 0.833·25-s − 0.192·27-s + 0.472·29-s + 0.873·31-s + 0.955·33-s − 0.320·35-s − 1.14·37-s − 0.173·39-s + 0.529·41-s − 0.591·43-s − 0.135·45-s − 0.148·47-s − 0.381·49-s − 0.691·51-s + 0.896·53-s + 0.674·55-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: \(1\)
Selberg data: \((2,\ 24,\ (\ :21/2),\ -1)\)

Particular Values

\(L(11)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{23}{2})\) 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 \( 1 + 8.90e6T + 4.76e14T^{2} \)
7 \( 1 - 5.87e8T + 5.58e17T^{2} \)
11 \( 1 + 1.42e11T + 7.40e21T^{2} \)
13 \( 1 - 1.49e11T + 2.47e23T^{2} \)
17 \( 1 - 9.95e12T + 6.90e25T^{2} \)
19 \( 1 + 5.02e12T + 7.14e26T^{2} \)
23 \( 1 - 3.25e14T + 3.94e28T^{2} \)
29 \( 1 - 1.07e15T + 5.13e30T^{2} \)
31 \( 1 - 3.98e15T + 2.08e31T^{2} \)
37 \( 1 + 3.35e16T + 8.55e32T^{2} \)
41 \( 1 - 4.54e16T + 7.38e33T^{2} \)
43 \( 1 + 8.38e16T + 2.00e34T^{2} \)
47 \( 1 + 5.34e16T + 1.30e35T^{2} \)
53 \( 1 - 1.14e18T + 1.62e36T^{2} \)
59 \( 1 + 5.97e18T + 1.54e37T^{2} \)
61 \( 1 - 4.12e17T + 3.10e37T^{2} \)
67 \( 1 + 2.42e19T + 2.22e38T^{2} \)
71 \( 1 - 7.25e16T + 7.52e38T^{2} \)
73 \( 1 + 2.97e19T + 1.34e39T^{2} \)
79 \( 1 + 5.78e19T + 7.08e39T^{2} \)
83 \( 1 + 2.48e20T + 1.99e40T^{2} \)
89 \( 1 + 5.56e19T + 8.65e40T^{2} \)
97 \( 1 + 8.39e20T + 5.27e41T^{2} \)
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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

−12.30182089701715984222073848654, −11.14965760894879747932707324007, −10.19236814942041903214563587048, −8.335302467683841981650666925753, −7.35235790111697782128860285239, −5.62189189686237243091434944827, −4.67388598607778752089276834617, −2.98258108472165282471093207779, −1.32041598908433861612565077638, 0, 1.32041598908433861612565077638, 2.98258108472165282471093207779, 4.67388598607778752089276834617, 5.62189189686237243091434944827, 7.35235790111697782128860285239, 8.335302467683841981650666925753, 10.19236814942041903214563587048, 11.14965760894879747932707324007, 12.30182089701715984222073848654

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