Properties

Label 2-24e2-576.13-c1-0-51
Degree $2$
Conductor $576$
Sign $-0.247 + 0.968i$
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  + (−0.864 − 1.11i)2-s + (0.0730 − 1.73i)3-s + (−0.504 + 1.93i)4-s + (−0.545 + 1.60i)5-s + (−1.99 + 1.41i)6-s + (1.46 − 1.90i)7-s + (2.60 − 1.10i)8-s + (−2.98 − 0.252i)9-s + (2.27 − 0.779i)10-s + (6.33 + 0.415i)11-s + (3.31 + 1.01i)12-s + (−1.78 + 2.03i)13-s + (−3.40 + 0.0114i)14-s + (2.74 + 1.06i)15-s + (−3.49 − 1.95i)16-s + (2.76 + 2.76i)17-s + ⋯
L(s)  = 1  + (−0.611 − 0.791i)2-s + (0.0421 − 0.999i)3-s + (−0.252 + 0.967i)4-s + (−0.244 + 0.719i)5-s + (−0.816 + 0.577i)6-s + (0.553 − 0.721i)7-s + (0.919 − 0.391i)8-s + (−0.996 − 0.0842i)9-s + (0.718 − 0.246i)10-s + (1.91 + 0.125i)11-s + (0.956 + 0.292i)12-s + (−0.495 + 0.564i)13-s + (−0.909 + 0.00305i)14-s + (0.708 + 0.274i)15-s + (−0.872 − 0.488i)16-s + (0.670 + 0.670i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.247 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.247 + 0.968i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.247 + 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.695672 - 0.895327i\)
\(L(\frac12)\) \(\approx\) \(0.695672 - 0.895327i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.864 + 1.11i)T \)
3 \( 1 + (-0.0730 + 1.73i)T \)
good5 \( 1 + (0.545 - 1.60i)T + (-3.96 - 3.04i)T^{2} \)
7 \( 1 + (-1.46 + 1.90i)T + (-1.81 - 6.76i)T^{2} \)
11 \( 1 + (-6.33 - 0.415i)T + (10.9 + 1.43i)T^{2} \)
13 \( 1 + (1.78 - 2.03i)T + (-1.69 - 12.8i)T^{2} \)
17 \( 1 + (-2.76 - 2.76i)T + 17iT^{2} \)
19 \( 1 + (-1.19 + 5.98i)T + (-17.5 - 7.27i)T^{2} \)
23 \( 1 + (-1.61 + 1.23i)T + (5.95 - 22.2i)T^{2} \)
29 \( 1 + (-3.38 + 6.86i)T + (-17.6 - 23.0i)T^{2} \)
31 \( 1 + (6.62 + 3.82i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.71 + 8.61i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (-3.28 + 2.51i)T + (10.6 - 39.6i)T^{2} \)
43 \( 1 + (0.389 - 5.93i)T + (-42.6 - 5.61i)T^{2} \)
47 \( 1 + (-1.18 + 4.40i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.39 - 3.58i)T + (-20.2 - 48.9i)T^{2} \)
59 \( 1 + (2.77 - 8.16i)T + (-46.8 - 35.9i)T^{2} \)
61 \( 1 + (9.29 + 4.58i)T + (37.1 + 48.3i)T^{2} \)
67 \( 1 + (0.821 + 12.5i)T + (-66.4 + 8.74i)T^{2} \)
71 \( 1 + (-1.14 - 2.76i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (3.73 - 9.00i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-13.9 - 3.75i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (2.92 - 0.993i)T + (65.8 - 50.5i)T^{2} \)
89 \( 1 + (-14.7 + 6.12i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + (-7.37 + 4.25i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.80933079640022450152339933619, −9.450889278072787145442635034528, −8.876109407836263126444759391015, −7.60519470694088222879906176920, −7.22073665774169019664034693162, −6.33125133440229631407614101495, −4.44601328595431587654503170123, −3.46053319981183233281139144931, −2.14213285125234468103660501149, −0.968590256972723501994044318099, 1.34528635897644016452736657676, 3.47660883477632088893657596431, 4.78360372318514712479724949268, 5.32962958879223950502304236612, 6.36441321032857145350872823121, 7.65757933923504760350027012886, 8.626612685751753945200313296224, 9.031013720758380598368534317974, 9.808581561615320184841579760754, 10.72258116873072719302214327642

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