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Properties

Label 2-24e2-48.35-c3-0-1
Degree $2$
Conductor $576$
Sign $0.670 - 0.741i$
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  + (−15.2 − 15.2i)5-s − 24.4·7-s + (−20.1 + 20.1i)11-s + (−26.7 − 26.7i)13-s − 85.9i·17-s + (−53.2 + 53.2i)19-s − 119. i·23-s + 337. i·25-s + (−78.5 + 78.5i)29-s − 200. i·31-s + (372. + 372. i)35-s + (−76.9 + 76.9i)37-s + 279.·41-s + (−15.7 − 15.7i)43-s + 470.·47-s + ⋯
L(s)  = 1  + (−1.36 − 1.36i)5-s − 1.32·7-s + (−0.553 + 0.553i)11-s + (−0.571 − 0.571i)13-s − 1.22i·17-s + (−0.643 + 0.643i)19-s − 1.08i·23-s + 2.69i·25-s + (−0.502 + 0.502i)29-s − 1.16i·31-s + (1.79 + 1.79i)35-s + (−0.342 + 0.342i)37-s + 1.06·41-s + (−0.0559 − 0.0559i)43-s + 1.46·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.670 - 0.741i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.670 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.670 - 0.741i$
Analytic conductor: \(33.9851\)
Root analytic conductor: \(5.82967\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :3/2),\ 0.670 - 0.741i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2418618188\)
\(L(\frac12)\) \(\approx\) \(0.2418618188\)
\(L(\frac{5}{2})\) 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 \( 1 + (15.2 + 15.2i)T + 125iT^{2} \)
7 \( 1 + 24.4T + 343T^{2} \)
11 \( 1 + (20.1 - 20.1i)T - 1.33e3iT^{2} \)
13 \( 1 + (26.7 + 26.7i)T + 2.19e3iT^{2} \)
17 \( 1 + 85.9iT - 4.91e3T^{2} \)
19 \( 1 + (53.2 - 53.2i)T - 6.85e3iT^{2} \)
23 \( 1 + 119. iT - 1.21e4T^{2} \)
29 \( 1 + (78.5 - 78.5i)T - 2.43e4iT^{2} \)
31 \( 1 + 200. iT - 2.97e4T^{2} \)
37 \( 1 + (76.9 - 76.9i)T - 5.06e4iT^{2} \)
41 \( 1 - 279.T + 6.89e4T^{2} \)
43 \( 1 + (15.7 + 15.7i)T + 7.95e4iT^{2} \)
47 \( 1 - 470.T + 1.03e5T^{2} \)
53 \( 1 + (-112. - 112. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-241. + 241. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-6.00 - 6.00i)T + 2.26e5iT^{2} \)
67 \( 1 + (273. - 273. i)T - 3.00e5iT^{2} \)
71 \( 1 - 448. iT - 3.57e5T^{2} \)
73 \( 1 - 54.7iT - 3.89e5T^{2} \)
79 \( 1 - 29.1iT - 4.93e5T^{2} \)
83 \( 1 + (893. + 893. i)T + 5.71e5iT^{2} \)
89 \( 1 - 281.T + 7.04e5T^{2} \)
97 \( 1 - 188.T + 9.12e5T^{2} \)
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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

−10.31364929351014630979392664267, −9.468375357434269394251143190483, −8.697407105651178284264168481128, −7.74228008171047028780409467477, −7.10098077913720571590758755107, −5.69917603211119208777778798377, −4.70814299387664174820998206055, −3.88333761830953838850625399985, −2.65834124152886950441926066782, −0.62509571375518097049373961525, 0.12472623056435361808966731929, 2.54367314223682903580996740008, 3.43083846289040205503136208903, 4.15469453671033324359973177395, 5.85648145599150862895778434560, 6.76021570418376484773962787537, 7.35068534820913735803737774068, 8.305662498824133289916459591580, 9.389127789979834030081057792287, 10.47980436520962295300716113330

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