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Properties

Label 2-24e2-144.85-c1-0-16
Degree $2$
Conductor $576$
Sign $0.542 + 0.840i$
Analytic cond. $4.59938$
Root an. cond. $2.14461$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 − 1.21i)3-s + (0.174 − 0.0468i)5-s + (4.04 − 2.33i)7-s + (0.0241 − 2.99i)9-s + (−0.160 + 0.598i)11-s + (1.18 + 4.41i)13-s + (0.157 − 0.270i)15-s − 4.34·17-s + (1.23 − 1.23i)19-s + (2.12 − 7.81i)21-s + (−3.86 − 2.23i)23-s + (−4.30 + 2.48i)25-s + (−3.62 − 3.71i)27-s + (8.64 + 2.31i)29-s + (2.25 − 3.90i)31-s + ⋯
L(s)  = 1  + (0.709 − 0.704i)3-s + (0.0781 − 0.0209i)5-s + (1.52 − 0.883i)7-s + (0.00806 − 0.999i)9-s + (−0.0483 + 0.180i)11-s + (0.327 + 1.22i)13-s + (0.0407 − 0.0698i)15-s − 1.05·17-s + (0.284 − 0.284i)19-s + (0.464 − 1.70i)21-s + (−0.805 − 0.465i)23-s + (−0.860 + 0.496i)25-s + (−0.698 − 0.715i)27-s + (1.60 + 0.430i)29-s + (0.404 − 0.701i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 + 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.542 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.542 + 0.840i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.542 + 0.840i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85192 - 1.00876i\)
\(L(\frac12)\) \(\approx\) \(1.85192 - 1.00876i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.22 + 1.21i)T \)
good5 \( 1 + (-0.174 + 0.0468i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-4.04 + 2.33i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.160 - 0.598i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.18 - 4.41i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + 4.34T + 17T^{2} \)
19 \( 1 + (-1.23 + 1.23i)T - 19iT^{2} \)
23 \( 1 + (3.86 + 2.23i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-8.64 - 2.31i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-2.25 + 3.90i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.79 - 2.79i)T + 37iT^{2} \)
41 \( 1 + (3.67 + 2.12i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.00351 - 0.0131i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-1.17 - 2.03i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.519 + 0.519i)T + 53iT^{2} \)
59 \( 1 + (11.0 - 2.95i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.19 + 0.588i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-1.88 - 7.04i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 7.55iT - 71T^{2} \)
73 \( 1 + 2.92iT - 73T^{2} \)
79 \( 1 + (1.45 + 2.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.44 - 1.99i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 3.18iT - 89T^{2} \)
97 \( 1 + (-8.03 - 13.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.68790753713964146308955783384, −9.563457520896279782014384631376, −8.610949945616374263547955300279, −7.973239078783534658613281912100, −7.11620890849003425686191459413, −6.31564651060418424877317720352, −4.70019781286215947439767774299, −4.00827639087892154712839668214, −2.31472514236449186914427854126, −1.33222207436237499550922925360, 1.89268249897318812945926726318, 2.96973524441486909516399305186, 4.34088671208280753236704978159, 5.14486839520708863690509244774, 6.08371957005769520963119849171, 7.83390932997559041505784705419, 8.207590314950983004801304946588, 8.948596441580282913496623145743, 10.02737605348225116439980058192, 10.77718308549629751972151459043

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