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Properties

Label 2-24-8.5-c7-0-11
Degree $2$
Conductor $24$
Sign $-0.680 + 0.732i$
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  + (3.06 + 10.8i)2-s − 27i·3-s + (−109. + 66.7i)4-s − 23.0i·5-s + (294. − 82.6i)6-s − 1.54e3·7-s + (−1.06e3 − 985. i)8-s − 729·9-s + (250. − 70.5i)10-s − 822. i·11-s + (1.80e3 + 2.94e3i)12-s − 5.92e3i·13-s + (−4.74e3 − 1.68e4i)14-s − 621.·15-s + (7.48e3 − 1.45e4i)16-s − 1.27e4·17-s + ⋯
L(s)  = 1  + (0.270 + 0.962i)2-s − 0.577i·3-s + (−0.853 + 0.521i)4-s − 0.0823i·5-s + (0.555 − 0.156i)6-s − 1.70·7-s + (−0.732 − 0.680i)8-s − 0.333·9-s + (0.0792 − 0.0222i)10-s − 0.186i·11-s + (0.300 + 0.492i)12-s − 0.748i·13-s + (−0.461 − 1.64i)14-s − 0.0475·15-s + (0.456 − 0.889i)16-s − 0.628·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.680 + 0.732i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.680 + 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24\)    =    \(2^{3} \cdot 3\)
Sign: $-0.680 + 0.732i$
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: \((2,\ 24,\ (\ :7/2),\ -0.680 + 0.732i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.0189467 - 0.0434471i\)
\(L(\frac12)\) \(\approx\) \(0.0189467 - 0.0434471i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.06 - 10.8i)T \)
3 \( 1 + 27iT \)
good5 \( 1 + 23.0iT - 7.81e4T^{2} \)
7 \( 1 + 1.54e3T + 8.23e5T^{2} \)
11 \( 1 + 822. iT - 1.94e7T^{2} \)
13 \( 1 + 5.92e3iT - 6.27e7T^{2} \)
17 \( 1 + 1.27e4T + 4.10e8T^{2} \)
19 \( 1 - 4.38e4iT - 8.93e8T^{2} \)
23 \( 1 + 7.76e4T + 3.40e9T^{2} \)
29 \( 1 - 1.51e5iT - 1.72e10T^{2} \)
31 \( 1 + 4.64e4T + 2.75e10T^{2} \)
37 \( 1 - 3.24e4iT - 9.49e10T^{2} \)
41 \( 1 - 1.88e5T + 1.94e11T^{2} \)
43 \( 1 + 8.47e5iT - 2.71e11T^{2} \)
47 \( 1 + 1.23e6T + 5.06e11T^{2} \)
53 \( 1 + 1.00e6iT - 1.17e12T^{2} \)
59 \( 1 + 2.08e6iT - 2.48e12T^{2} \)
61 \( 1 + 2.52e5iT - 3.14e12T^{2} \)
67 \( 1 - 4.04e6iT - 6.06e12T^{2} \)
71 \( 1 + 2.06e6T + 9.09e12T^{2} \)
73 \( 1 - 1.18e6T + 1.10e13T^{2} \)
79 \( 1 + 3.44e6T + 1.92e13T^{2} \)
83 \( 1 + 4.07e6iT - 2.71e13T^{2} \)
89 \( 1 + 1.21e7T + 4.42e13T^{2} \)
97 \( 1 + 1.02e7T + 8.07e13T^{2} \)
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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

−15.87953801681397884866674440419, −14.37980947811087056190405600224, −13.09056792641660803709929592969, −12.42699555978874903757305234074, −9.974742568623782439922583894632, −8.462750407773251849219431026993, −6.90830086528985489477965007360, −5.80691226548244380284334648138, −3.43479182876951135247255595750, −0.02189198608018449540391208970, 2.79094716188246887088337817399, 4.32934517007049713820822945505, 6.34327163698082863186145371189, 9.110945324410222977356970092875, 9.939503085758219003451337201268, 11.31900398254390550850274619264, 12.72680759039857000808259157581, 13.76484885835742512663245278753, 15.29893053087352513494430588173, 16.46910536063118408040049772026

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