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Properties

Label 2-24-8.5-c7-0-10
Degree $2$
Conductor $24$
Sign $0.978 + 0.207i$
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  + (9.38 + 6.32i)2-s − 27i·3-s + (48.0 + 118. i)4-s − 425. i·5-s + (170. − 253. i)6-s + 1.66e3·7-s + (−299. + 1.41e3i)8-s − 729·9-s + (2.68e3 − 3.98e3i)10-s − 2.46e3i·11-s + (3.20e3 − 1.29e3i)12-s + 3.76e3i·13-s + (1.56e4 + 1.05e4i)14-s − 1.14e4·15-s + (−1.17e4 + 1.13e4i)16-s + 1.62e4·17-s + ⋯
L(s)  = 1  + (0.829 + 0.558i)2-s − 0.577i·3-s + (0.375 + 0.926i)4-s − 1.52i·5-s + (0.322 − 0.478i)6-s + 1.83·7-s + (−0.207 + 0.978i)8-s − 0.333·9-s + (0.850 − 1.26i)10-s − 0.558i·11-s + (0.535 − 0.216i)12-s + 0.475i·13-s + (1.52 + 1.02i)14-s − 0.878·15-s + (−0.718 + 0.695i)16-s + 0.801·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24\)    =    \(2^{3} \cdot 3\)
Sign: $0.978 + 0.207i$
Analytic conductor: \(7.49724\)
Root analytic conductor: \(2.73810\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{24} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 24,\ (\ :7/2),\ 0.978 + 0.207i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.85817 - 0.299283i\)
\(L(\frac12)\) \(\approx\) \(2.85817 - 0.299283i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-9.38 - 6.32i)T \)
3 \( 1 + 27iT \)
good5 \( 1 + 425. iT - 7.81e4T^{2} \)
7 \( 1 - 1.66e3T + 8.23e5T^{2} \)
11 \( 1 + 2.46e3iT - 1.94e7T^{2} \)
13 \( 1 - 3.76e3iT - 6.27e7T^{2} \)
17 \( 1 - 1.62e4T + 4.10e8T^{2} \)
19 \( 1 + 4.86e3iT - 8.93e8T^{2} \)
23 \( 1 + 1.08e5T + 3.40e9T^{2} \)
29 \( 1 - 8.90e4iT - 1.72e10T^{2} \)
31 \( 1 + 6.94e4T + 2.75e10T^{2} \)
37 \( 1 - 4.18e5iT - 9.49e10T^{2} \)
41 \( 1 - 2.74e5T + 1.94e11T^{2} \)
43 \( 1 - 4.62e5iT - 2.71e11T^{2} \)
47 \( 1 - 1.53e5T + 5.06e11T^{2} \)
53 \( 1 + 1.81e5iT - 1.17e12T^{2} \)
59 \( 1 + 6.47e5iT - 2.48e12T^{2} \)
61 \( 1 - 2.26e6iT - 3.14e12T^{2} \)
67 \( 1 + 2.31e6iT - 6.06e12T^{2} \)
71 \( 1 - 2.77e6T + 9.09e12T^{2} \)
73 \( 1 + 4.45e6T + 1.10e13T^{2} \)
79 \( 1 + 3.14e6T + 1.92e13T^{2} \)
83 \( 1 + 3.33e6iT - 2.71e13T^{2} \)
89 \( 1 + 5.00e6T + 4.42e13T^{2} \)
97 \( 1 - 8.06e6T + 8.07e13T^{2} \)
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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

−16.18762699626850635452168032896, −14.53711639634887441635061423925, −13.66559353329441767835478612109, −12.32193088299631047555176989311, −11.50845665167843592736727641150, −8.572984809219119078922999320530, −7.82441753465044894300436511613, −5.63432732948343729085552111774, −4.49135763595307966593538620898, −1.55093232005834510317091209817, 2.19162229467240421196006453914, 4.00446098614917634854327719226, 5.64646239247774241155275159333, 7.61967390169276163768674539049, 10.12026029996996216219315018409, 10.95843691686261131509617007788, 11.96268467548298983693834973024, 14.18710410158584405199051861915, 14.54522580529395700587612514395, 15.56146522107347737000676907879

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