Properties

Label 2-24e2-576.517-c1-0-72
Degree $2$
Conductor $576$
Sign $0.994 - 0.104i$
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  + (1.30 + 0.543i)2-s + (1.15 − 1.28i)3-s + (1.40 + 1.42i)4-s + (−0.183 + 0.209i)5-s + (2.21 − 1.05i)6-s + (1.21 − 0.159i)7-s + (1.06 + 2.61i)8-s + (−0.319 − 2.98i)9-s + (−0.354 + 0.173i)10-s + (−0.510 + 1.03i)11-s + (3.45 − 0.169i)12-s + (−0.0481 + 0.141i)13-s + (1.67 + 0.451i)14-s + (0.0572 + 0.479i)15-s + (−0.0334 + 3.99i)16-s + (2.67 − 2.67i)17-s + ⋯
L(s)  = 1  + (0.923 + 0.384i)2-s + (0.668 − 0.743i)3-s + (0.704 + 0.710i)4-s + (−0.0822 + 0.0937i)5-s + (0.903 − 0.429i)6-s + (0.458 − 0.0603i)7-s + (0.376 + 0.926i)8-s + (−0.106 − 0.994i)9-s + (−0.111 + 0.0549i)10-s + (−0.153 + 0.311i)11-s + (0.998 − 0.0490i)12-s + (−0.0133 + 0.0393i)13-s + (0.446 + 0.120i)14-s + (0.0147 + 0.123i)15-s + (−0.00837 + 0.999i)16-s + (0.648 − 0.648i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.104i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.994 - 0.104i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.994 - 0.104i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.15240 + 0.165508i\)
\(L(\frac12)\) \(\approx\) \(3.15240 + 0.165508i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.30 - 0.543i)T \)
3 \( 1 + (-1.15 + 1.28i)T \)
good5 \( 1 + (0.183 - 0.209i)T + (-0.652 - 4.95i)T^{2} \)
7 \( 1 + (-1.21 + 0.159i)T + (6.76 - 1.81i)T^{2} \)
11 \( 1 + (0.510 - 1.03i)T + (-6.69 - 8.72i)T^{2} \)
13 \( 1 + (0.0481 - 0.141i)T + (-10.3 - 7.91i)T^{2} \)
17 \( 1 + (-2.67 + 2.67i)T - 17iT^{2} \)
19 \( 1 + (-0.709 - 3.56i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (-0.854 + 6.49i)T + (-22.2 - 5.95i)T^{2} \)
29 \( 1 + (5.01 + 0.328i)T + (28.7 + 3.78i)T^{2} \)
31 \( 1 + (7.19 + 4.15i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.345 + 1.73i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (0.636 - 4.83i)T + (-39.6 - 10.6i)T^{2} \)
43 \( 1 + (-5.19 - 2.56i)T + (26.1 + 34.1i)T^{2} \)
47 \( 1 + (7.62 + 2.04i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.478 + 0.715i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (-5.66 + 6.45i)T + (-7.70 - 58.4i)T^{2} \)
61 \( 1 + (0.723 - 11.0i)T + (-60.4 - 7.96i)T^{2} \)
67 \( 1 + (-1.14 + 0.565i)T + (40.7 - 53.1i)T^{2} \)
71 \( 1 + (1.43 - 3.46i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (1.96 + 4.73i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (0.0723 - 0.270i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (-0.835 + 0.732i)T + (10.8 - 82.2i)T^{2} \)
89 \( 1 + (16.2 + 6.74i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (-7.35 + 4.24i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.11049970599590762426033754782, −9.771786890014518626174731667920, −8.671454146017057762234407174723, −7.71839126343814919435770345282, −7.29751353189576914661472486480, −6.21656673298194847891769751321, −5.22934520617215579211388344219, −3.99976204458748558972315018531, −2.97878735095466047686392392532, −1.78377202963863188829882691106, 1.79960525672504898841649150419, 3.11327955945123441352539304499, 3.92437021363297209602976155147, 5.01527275590618113833419180597, 5.68025485451800137090336985836, 7.13763195684494315062125810424, 8.067295761106794627500580325911, 9.130364967757855934840606585418, 9.966830099626529807277160456489, 10.88037788064837867965884399398

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