Properties

Label 2-24-24.11-c3-0-3
Degree $2$
Conductor $24$
Sign $0.272 - 0.962i$
Analytic cond. $1.41604$
Root an. cond. $1.18997$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82i·2-s + (5 + 1.41i)3-s − 8.00·4-s + (−4.00 + 14.1i)6-s − 22.6i·8-s + (23 + 14.1i)9-s − 70.7i·11-s + (−40.0 − 11.3i)12-s + 64.0·16-s + 107. i·17-s + (−40.0 + 65.0i)18-s − 106·19-s + 200.·22-s + (32.0 − 113. i)24-s − 125·25-s + ⋯
L(s)  = 1  + 0.999i·2-s + (0.962 + 0.272i)3-s − 1.00·4-s + (−0.272 + 0.962i)6-s − 1.00i·8-s + (0.851 + 0.523i)9-s − 1.93i·11-s + (−0.962 − 0.272i)12-s + 1.00·16-s + 1.53i·17-s + (−0.523 + 0.851i)18-s − 1.27·19-s + 1.93·22-s + (0.272 − 0.962i)24-s − 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.272 - 0.962i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.272 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24\)    =    \(2^{3} \cdot 3\)
Sign: $0.272 - 0.962i$
Analytic conductor: \(1.41604\)
Root analytic conductor: \(1.18997\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{24} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 24,\ (\ :3/2),\ 0.272 - 0.962i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.05837 + 0.800543i\)
\(L(\frac12)\) \(\approx\) \(1.05837 + 0.800543i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.82iT \)
3 \( 1 + (-5 - 1.41i)T \)
good5 \( 1 + 125T^{2} \)
7 \( 1 - 343T^{2} \)
11 \( 1 + 70.7iT - 1.33e3T^{2} \)
13 \( 1 - 2.19e3T^{2} \)
17 \( 1 - 107. iT - 4.91e3T^{2} \)
19 \( 1 + 106T + 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 + 2.43e4T^{2} \)
31 \( 1 - 2.97e4T^{2} \)
37 \( 1 - 5.06e4T^{2} \)
41 \( 1 - 56.5iT - 6.89e4T^{2} \)
43 \( 1 - 290T + 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 + 1.48e5T^{2} \)
59 \( 1 + 325. iT - 2.05e5T^{2} \)
61 \( 1 - 2.26e5T^{2} \)
67 \( 1 + 70T + 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 + 430T + 3.89e5T^{2} \)
79 \( 1 - 4.93e5T^{2} \)
83 \( 1 + 681. iT - 5.71e5T^{2} \)
89 \( 1 - 1.32e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.91e3T + 9.12e5T^{2} \)
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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

−17.17707662552019832708411484596, −16.09487684216355360453776254408, −15.02070220758083529490572384490, −13.97190630998819148280329432401, −12.99333616926732856934058342174, −10.54909112750273714796651660759, −8.873267286634214673132337974018, −8.045852476982021650881049614039, −6.07442499956795941981547089926, −3.85775137198782991165286996682, 2.26027663966447729927884215960, 4.37526749996133579850317243106, 7.44201540410787076520053969815, 9.108404416454209027254155882530, 10.12289544376962637171298744557, 12.01523309172515279372958118806, 13.01653052415009321306624471507, 14.23141439033694605645255625899, 15.34802142870530138950796854847, 17.49537251847867538030445534778

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