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Properties

Label 2-24e2-8.5-c3-0-2
Degree $2$
Conductor $576$
Sign $-0.965 + 0.258i$
Analytic cond. $33.9851$
Root an. cond. $5.82967$
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  + 13.8i·5-s − 3.46·7-s + 48i·11-s − 20.7i·13-s − 96·17-s + 40i·19-s + 110.·23-s − 66.9·25-s + 13.8i·29-s − 204.·31-s − 47.9i·35-s − 297. i·37-s + 288·41-s + 152i·43-s − 554.·47-s + ⋯
L(s)  = 1  + 1.23i·5-s − 0.187·7-s + 1.31i·11-s − 0.443i·13-s − 1.36·17-s + 0.482i·19-s + 1.00·23-s − 0.535·25-s + 0.0887i·29-s − 1.18·31-s − 0.231i·35-s − 1.32i·37-s + 1.09·41-s + 0.539i·43-s − 1.72·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.965 + 0.258i$
Analytic conductor: \(33.9851\)
Root analytic conductor: \(5.82967\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :3/2),\ -0.965 + 0.258i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5897749124\)
\(L(\frac12)\) \(\approx\) \(0.5897749124\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 - 13.8iT - 125T^{2} \)
7 \( 1 + 3.46T + 343T^{2} \)
11 \( 1 - 48iT - 1.33e3T^{2} \)
13 \( 1 + 20.7iT - 2.19e3T^{2} \)
17 \( 1 + 96T + 4.91e3T^{2} \)
19 \( 1 - 40iT - 6.85e3T^{2} \)
23 \( 1 - 110.T + 1.21e4T^{2} \)
29 \( 1 - 13.8iT - 2.43e4T^{2} \)
31 \( 1 + 204.T + 2.97e4T^{2} \)
37 \( 1 + 297. iT - 5.06e4T^{2} \)
41 \( 1 - 288T + 6.89e4T^{2} \)
43 \( 1 - 152iT - 7.95e4T^{2} \)
47 \( 1 + 554.T + 1.03e5T^{2} \)
53 \( 1 + 180. iT - 1.48e5T^{2} \)
59 \( 1 + 480iT - 2.05e5T^{2} \)
61 \( 1 + 755. iT - 2.26e5T^{2} \)
67 \( 1 - 848iT - 3.00e5T^{2} \)
71 \( 1 - 886.T + 3.57e5T^{2} \)
73 \( 1 + 538T + 3.89e5T^{2} \)
79 \( 1 + 1.00e3T + 4.93e5T^{2} \)
83 \( 1 - 432iT - 5.71e5T^{2} \)
89 \( 1 + 1.34e3T + 7.04e5T^{2} \)
97 \( 1 + 590T + 9.12e5T^{2} \)
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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

−10.92283797976619360680728349558, −9.933945562282452087185933952145, −9.246294415014027579261941866150, −7.975654751871508277088270827907, −7.02991556412982638618126072465, −6.57421767281394597707947129505, −5.26892789920994912137480514537, −4.10152816305759769137667084923, −2.95417428465133995472283705268, −1.92878372782111158257271379353, 0.17231599488565185040479764797, 1.38799773224146188372363016184, 2.94771208442251610340594274755, 4.26322150389409097728763187995, 5.09457672981485053473622289358, 6.11639578024126330290441820489, 7.09932203002597565994470164810, 8.433397318144477148917388767415, 8.835836310582732595034376904638, 9.590474852056316444081428826730

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