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Properties

Label 2-24-24.11-c7-0-9
Degree $2$
Conductor $24$
Sign $0.489 + 0.871i$
Analytic cond. $7.49724$
Root an. cond. $2.73810$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−10.0 − 5.27i)2-s + (−43.1 + 18.0i)3-s + (72.3 + 105. i)4-s − 277.·5-s + (527. + 46.7i)6-s + 1.06e3i·7-s + (−166. − 1.43e3i)8-s + (1.53e3 − 1.55e3i)9-s + (2.77e3 + 1.46e3i)10-s − 5.37e3i·11-s + (−5.02e3 − 3.24e3i)12-s − 7.99e3i·13-s + (5.62e3 − 1.06e4i)14-s + (1.19e4 − 5.01e3i)15-s + (−5.91e3 + 1.52e4i)16-s + 1.62e4i·17-s + ⋯
L(s)  = 1  + (−0.884 − 0.466i)2-s + (−0.922 + 0.386i)3-s + (0.565 + 0.824i)4-s − 0.993·5-s + (0.996 + 0.0884i)6-s + 1.17i·7-s + (−0.115 − 0.993i)8-s + (0.701 − 0.712i)9-s + (0.878 + 0.463i)10-s − 1.21i·11-s + (−0.839 − 0.542i)12-s − 1.00i·13-s + (0.547 − 1.03i)14-s + (0.916 − 0.383i)15-s + (−0.361 + 0.932i)16-s + 0.804i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24\)    =    \(2^{3} \cdot 3\)
Sign: $0.489 + 0.871i$
Analytic conductor: \(7.49724\)
Root analytic conductor: \(2.73810\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{24} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 24,\ (\ :7/2),\ 0.489 + 0.871i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.453829 - 0.265524i\)
\(L(\frac12)\) \(\approx\) \(0.453829 - 0.265524i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (10.0 + 5.27i)T \)
3 \( 1 + (43.1 - 18.0i)T \)
good5 \( 1 + 277.T + 7.81e4T^{2} \)
7 \( 1 - 1.06e3iT - 8.23e5T^{2} \)
11 \( 1 + 5.37e3iT - 1.94e7T^{2} \)
13 \( 1 + 7.99e3iT - 6.27e7T^{2} \)
17 \( 1 - 1.62e4iT - 4.10e8T^{2} \)
19 \( 1 - 4.59e4T + 8.93e8T^{2} \)
23 \( 1 - 5.19e4T + 3.40e9T^{2} \)
29 \( 1 + 2.53e4T + 1.72e10T^{2} \)
31 \( 1 + 1.68e5iT - 2.75e10T^{2} \)
37 \( 1 + 2.40e5iT - 9.49e10T^{2} \)
41 \( 1 + 2.47e5iT - 1.94e11T^{2} \)
43 \( 1 - 2.88e5T + 2.71e11T^{2} \)
47 \( 1 - 8.95e5T + 5.06e11T^{2} \)
53 \( 1 + 2.05e6T + 1.17e12T^{2} \)
59 \( 1 + 5.74e5iT - 2.48e12T^{2} \)
61 \( 1 + 1.45e6iT - 3.14e12T^{2} \)
67 \( 1 + 2.09e5T + 6.06e12T^{2} \)
71 \( 1 + 6.22e5T + 9.09e12T^{2} \)
73 \( 1 - 4.04e6T + 1.10e13T^{2} \)
79 \( 1 + 7.77e6iT - 1.92e13T^{2} \)
83 \( 1 - 3.75e6iT - 2.71e13T^{2} \)
89 \( 1 + 3.64e6iT - 4.42e13T^{2} \)
97 \( 1 - 2.20e5T + 8.07e13T^{2} \)
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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

−15.95473242240639414361578468092, −15.41480345916275216240100519697, −12.65278787621293299317947849000, −11.66744863269494683255061613280, −10.84156324804361103769081822079, −9.200010116496041565149057295691, −7.80377713153088578417764073350, −5.78149131144123330973099820625, −3.40068838739925804450315590456, −0.53738980062406917533260497473, 1.07624500878691080410163427946, 4.74418856775752654761228744159, 6.99666625959612085776621069480, 7.50338469346260848543346835616, 9.698444475455203289737068448083, 11.06370839595589999672306681090, 12.01870294702223724053560759307, 13.95116585295854035338340623050, 15.61653025860739270448772492133, 16.50333904918241762139397964288

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