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Properties

Label 2-24-24.11-c7-0-19
Degree $2$
Conductor $24$
Sign $-0.0382 + 0.999i$
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  + (−10.9 + 3.02i)2-s + (35.1 − 30.8i)3-s + (109. − 66.0i)4-s − 65.0·5-s + (−290. + 442. i)6-s + 90.3i·7-s + (−995. + 1.05e3i)8-s + (288. − 2.16e3i)9-s + (709. − 197. i)10-s − 3.67e3i·11-s + (1.82e3 − 5.70e3i)12-s − 1.24e4i·13-s + (−273. − 984. i)14-s + (−2.28e3 + 2.00e3i)15-s + (7.66e3 − 1.44e4i)16-s − 5.04e3i·17-s + ⋯
L(s)  = 1  + (−0.963 + 0.267i)2-s + (0.752 − 0.658i)3-s + (0.856 − 0.515i)4-s − 0.232·5-s + (−0.548 + 0.836i)6-s + 0.0995i·7-s + (−0.687 + 0.726i)8-s + (0.131 − 0.991i)9-s + (0.224 − 0.0623i)10-s − 0.832i·11-s + (0.304 − 0.952i)12-s − 1.57i·13-s + (−0.0266 − 0.0958i)14-s + (−0.175 + 0.153i)15-s + (0.467 − 0.883i)16-s − 0.249i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0382 + 0.999i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.0382 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24\)    =    \(2^{3} \cdot 3\)
Sign: $-0.0382 + 0.999i$
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: \((2,\ 24,\ (\ :7/2),\ -0.0382 + 0.999i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.785408 - 0.816065i\)
\(L(\frac12)\) \(\approx\) \(0.785408 - 0.816065i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (10.9 - 3.02i)T \)
3 \( 1 + (-35.1 + 30.8i)T \)
good5 \( 1 + 65.0T + 7.81e4T^{2} \)
7 \( 1 - 90.3iT - 8.23e5T^{2} \)
11 \( 1 + 3.67e3iT - 1.94e7T^{2} \)
13 \( 1 + 1.24e4iT - 6.27e7T^{2} \)
17 \( 1 + 5.04e3iT - 4.10e8T^{2} \)
19 \( 1 + 2.07e4T + 8.93e8T^{2} \)
23 \( 1 - 1.57e4T + 3.40e9T^{2} \)
29 \( 1 - 2.18e5T + 1.72e10T^{2} \)
31 \( 1 + 1.79e5iT - 2.75e10T^{2} \)
37 \( 1 - 4.76e5iT - 9.49e10T^{2} \)
41 \( 1 - 4.06e5iT - 1.94e11T^{2} \)
43 \( 1 - 6.51e5T + 2.71e11T^{2} \)
47 \( 1 + 9.39e5T + 5.06e11T^{2} \)
53 \( 1 - 1.25e6T + 1.17e12T^{2} \)
59 \( 1 - 5.39e5iT - 2.48e12T^{2} \)
61 \( 1 - 1.69e6iT - 3.14e12T^{2} \)
67 \( 1 - 1.56e6T + 6.06e12T^{2} \)
71 \( 1 + 2.62e6T + 9.09e12T^{2} \)
73 \( 1 - 1.17e6T + 1.10e13T^{2} \)
79 \( 1 + 1.68e6iT - 1.92e13T^{2} \)
83 \( 1 + 8.05e6iT - 2.71e13T^{2} \)
89 \( 1 + 7.67e6iT - 4.42e13T^{2} \)
97 \( 1 - 3.17e6T + 8.07e13T^{2} \)
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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

−15.75655817564974571075312526572, −14.82125689227913032002376976151, −13.31991484963329143805003616700, −11.77879187521699811264847608772, −10.18283450868295285336695035554, −8.612703612609082122215463979779, −7.75946339722032975118022301341, −6.14149824863597022955098280103, −2.82665064436670909528334520265, −0.75061245326410811034737973280, 2.11010534261430065198563618312, 4.09675558737080665159806360222, 7.05293210873911318065038700324, 8.538984293337736547984810834567, 9.615103199951453224958076055558, 10.82725892681529014713262657283, 12.31087726089060910522906828264, 14.15367146659394489447177029439, 15.47725183474142673119508999401, 16.41900128461609046329339779232

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