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Properties

Label 2-24e2-16.13-c3-0-1
Degree $2$
Conductor $576$
Sign $-0.833 - 0.551i$
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  + (9.19 − 9.19i)5-s + 31.8i·7-s + (−24.6 + 24.6i)11-s + (43.7 + 43.7i)13-s − 128.·17-s + (−36.8 − 36.8i)19-s − 97.8i·23-s − 44.1i·25-s + (−70.2 − 70.2i)29-s − 280.·31-s + (293. + 293. i)35-s + (−85.9 + 85.9i)37-s − 153. i·41-s + (−14.1 + 14.1i)43-s + 318.·47-s + ⋯
L(s)  = 1  + (0.822 − 0.822i)5-s + 1.72i·7-s + (−0.676 + 0.676i)11-s + (0.933 + 0.933i)13-s − 1.83·17-s + (−0.444 − 0.444i)19-s − 0.886i·23-s − 0.353i·25-s + (−0.449 − 0.449i)29-s − 1.62·31-s + (1.41 + 1.41i)35-s + (−0.382 + 0.382i)37-s − 0.583i·41-s + (−0.0502 + 0.0502i)43-s + 0.987·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.833 - 0.551i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.833 - 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.833 - 0.551i$
Analytic conductor: \(33.9851\)
Root analytic conductor: \(5.82967\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :3/2),\ -0.833 - 0.551i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9365463389\)
\(L(\frac12)\) \(\approx\) \(0.9365463389\)
\(L(\frac{5}{2})\) 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 \( 1 + (-9.19 + 9.19i)T - 125iT^{2} \)
7 \( 1 - 31.8iT - 343T^{2} \)
11 \( 1 + (24.6 - 24.6i)T - 1.33e3iT^{2} \)
13 \( 1 + (-43.7 - 43.7i)T + 2.19e3iT^{2} \)
17 \( 1 + 128.T + 4.91e3T^{2} \)
19 \( 1 + (36.8 + 36.8i)T + 6.85e3iT^{2} \)
23 \( 1 + 97.8iT - 1.21e4T^{2} \)
29 \( 1 + (70.2 + 70.2i)T + 2.43e4iT^{2} \)
31 \( 1 + 280.T + 2.97e4T^{2} \)
37 \( 1 + (85.9 - 85.9i)T - 5.06e4iT^{2} \)
41 \( 1 + 153. iT - 6.89e4T^{2} \)
43 \( 1 + (14.1 - 14.1i)T - 7.95e4iT^{2} \)
47 \( 1 - 318.T + 1.03e5T^{2} \)
53 \( 1 + (233. - 233. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-46.8 + 46.8i)T - 2.05e5iT^{2} \)
61 \( 1 + (36.7 + 36.7i)T + 2.26e5iT^{2} \)
67 \( 1 + (-122. - 122. i)T + 3.00e5iT^{2} \)
71 \( 1 - 268. iT - 3.57e5T^{2} \)
73 \( 1 - 774. iT - 3.89e5T^{2} \)
79 \( 1 + 30.1T + 4.93e5T^{2} \)
83 \( 1 + (294. + 294. i)T + 5.71e5iT^{2} \)
89 \( 1 - 156. iT - 7.04e5T^{2} \)
97 \( 1 - 485.T + 9.12e5T^{2} \)
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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

−10.75143751002728509437769948423, −9.413626357284870180138902857894, −8.978694487276565074516352616770, −8.461718463565912113700226378326, −6.89107029413590826124177321948, −5.99295925856097923965058943565, −5.21735902062457532845685815278, −4.29269112632492635992867708169, −2.39939852224381116731324683407, −1.86615124631906220999749332992, 0.24943302540183207156266049285, 1.76386404896364635325919520202, 3.18452609214920048418644077179, 4.05736282485764127876781699706, 5.47580044841364200468167317856, 6.39059397337915615646371648661, 7.18295017455935816466623610332, 8.069973628156804297394753345458, 9.174668308755412808374540786130, 10.27712450874474515989382516202

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