↑

Properties

Label 2-24e2-9.2-c2-0-34
Degree $2$
Conductor $576$
Sign $0.936 + 0.350i$
Analytic cond. $15.6948$
Root an. cond. $3.96167$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.44 + 1.73i)3-s + (4.5 − 2.59i)5-s + (4.17 − 7.22i)7-s + (2.99 + 8.48i)9-s + (−0.825 − 0.476i)11-s + (−4.84 − 8.39i)13-s + (15.5 + 1.43i)15-s − 18.8i·17-s + 24.6·19-s + (22.7 − 10.4i)21-s + (0.825 − 0.476i)23-s + (1 − 1.73i)25-s + (−7.34 + 25.9i)27-s + (−11.8 − 6.84i)29-s + (−1.52 − 2.63i)31-s + ⋯
L(s)  = 1  + (0.816 + 0.577i)3-s + (0.900 − 0.519i)5-s + (0.596 − 1.03i)7-s + (0.333 + 0.942i)9-s + (−0.0750 − 0.0433i)11-s + (−0.372 − 0.645i)13-s + (1.03 + 0.0953i)15-s − 1.11i·17-s + 1.29·19-s + (1.08 − 0.499i)21-s + (0.0359 − 0.0207i)23-s + (0.0400 − 0.0692i)25-s + (−0.272 + 0.962i)27-s + (−0.408 − 0.235i)29-s + (−0.0491 − 0.0850i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.936 + 0.350i$
Analytic conductor: \(15.6948\)
Root analytic conductor: \(3.96167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ 0.936 + 0.350i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.958883251\)
\(L(\frac12)\) \(\approx\) \(2.958883251\)
\(L(2)\) 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 \( 1 + (-2.44 - 1.73i)T \)
good5 \( 1 + (-4.5 + 2.59i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (-4.17 + 7.22i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (0.825 + 0.476i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (4.84 + 8.39i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 18.8iT - 289T^{2} \)
19 \( 1 - 24.6T + 361T^{2} \)
23 \( 1 + (-0.825 + 0.476i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (11.8 + 6.84i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (1.52 + 2.63i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 46.6T + 1.36e3T^{2} \)
41 \( 1 + (9.45 - 5.45i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-22.5 + 39.0i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-39.2 - 22.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 94.3iT - 2.80e3T^{2} \)
59 \( 1 + (-16.2 + 9.39i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-6.54 + 11.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-37.5 - 64.9i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 18.0iT - 5.04e3T^{2} \)
73 \( 1 + 7.90T + 5.32e3T^{2} \)
79 \( 1 + (-21.8 + 37.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-112. - 65.1i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 145. iT - 7.92e3T^{2} \)
97 \( 1 + (-54.9 + 95.1i)T + (-4.70e3 - 8.14e3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.27676858241449827806803555619, −9.581180601072097273509356552774, −8.915811705126446599029212471149, −7.74640871078992541026579699837, −7.24024755557370591752914599662, −5.49303056232982889356258580883, −4.91181907666153167758774024632, −3.74762076567690347426137012600, −2.52583929396175806632605174926, −1.14363641400829887948235354186, 1.68153121623067121179658239269, 2.36936354280385122870805063262, 3.57206568767037342698203742710, 5.16096268752917091409476465837, 6.09906734641613901182062933633, 6.98960994260469878653163488744, 7.975835891467460119709691851732, 8.842930554780331213729197558465, 9.526589116765043287222926522433, 10.39530649261724315897241620376

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