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Properties

Label 2-24e2-16.5-c3-0-21
Degree $2$
Conductor $576$
Sign $0.985 + 0.167i$
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  + (12.6 + 12.6i)5-s − 13.8i·7-s + (1.54 + 1.54i)11-s + (32.7 − 32.7i)13-s − 18.6·17-s + (86.4 − 86.4i)19-s − 134. i·23-s + 194. i·25-s + (59.7 − 59.7i)29-s + 31.5·31-s + (175. − 175. i)35-s + (89.1 + 89.1i)37-s + 210. i·41-s + (−119. − 119. i)43-s − 182.·47-s + ⋯
L(s)  = 1  + (1.13 + 1.13i)5-s − 0.749i·7-s + (0.0424 + 0.0424i)11-s + (0.699 − 0.699i)13-s − 0.266·17-s + (1.04 − 1.04i)19-s − 1.21i·23-s + 1.55i·25-s + (0.382 − 0.382i)29-s + 0.182·31-s + (0.847 − 0.847i)35-s + (0.396 + 0.396i)37-s + 0.801i·41-s + (−0.423 − 0.423i)43-s − 0.567·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.167i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.985 + 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.985 + 0.167i$
Analytic conductor: \(33.9851\)
Root analytic conductor: \(5.82967\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :3/2),\ 0.985 + 0.167i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.613572189\)
\(L(\frac12)\) \(\approx\) \(2.613572189\)
\(L(\frac{5}{2})\) 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 \( 1 + (-12.6 - 12.6i)T + 125iT^{2} \)
7 \( 1 + 13.8iT - 343T^{2} \)
11 \( 1 + (-1.54 - 1.54i)T + 1.33e3iT^{2} \)
13 \( 1 + (-32.7 + 32.7i)T - 2.19e3iT^{2} \)
17 \( 1 + 18.6T + 4.91e3T^{2} \)
19 \( 1 + (-86.4 + 86.4i)T - 6.85e3iT^{2} \)
23 \( 1 + 134. iT - 1.21e4T^{2} \)
29 \( 1 + (-59.7 + 59.7i)T - 2.43e4iT^{2} \)
31 \( 1 - 31.5T + 2.97e4T^{2} \)
37 \( 1 + (-89.1 - 89.1i)T + 5.06e4iT^{2} \)
41 \( 1 - 210. iT - 6.89e4T^{2} \)
43 \( 1 + (119. + 119. i)T + 7.95e4iT^{2} \)
47 \( 1 + 182.T + 1.03e5T^{2} \)
53 \( 1 + (-26.1 - 26.1i)T + 1.48e5iT^{2} \)
59 \( 1 + (-441. - 441. i)T + 2.05e5iT^{2} \)
61 \( 1 + (174. - 174. i)T - 2.26e5iT^{2} \)
67 \( 1 + (91.7 - 91.7i)T - 3.00e5iT^{2} \)
71 \( 1 - 348. iT - 3.57e5T^{2} \)
73 \( 1 + 299. iT - 3.89e5T^{2} \)
79 \( 1 - 943.T + 4.93e5T^{2} \)
83 \( 1 + (-313. + 313. i)T - 5.71e5iT^{2} \)
89 \( 1 + 1.41e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.51e3T + 9.12e5T^{2} \)
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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

−10.35378996345574066533575212421, −9.667987100343764239845601424385, −8.582431526562880975366279354628, −7.41172852982769507892578827303, −6.63895907087296686185736741146, −5.91645928923034707637540740845, −4.71115026910632670975477121929, −3.31709989167626574448445144906, −2.43336924552341363781255351242, −0.912778749431110011884182015611, 1.19680559789134606055581613003, 2.07519297875702675573573797471, 3.63673465990328815133221628172, 5.01169986666696186542008009636, 5.64748316596818043891481161183, 6.46153669280001320693936500756, 7.86781443894760645273215170301, 8.834953509233601283603749157838, 9.344261653028409004120162130289, 10.07495279175674569370917035530

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