Properties

Label 2-24e2-576.133-c1-0-74
Degree $2$
Conductor $576$
Sign $-0.0220 + 0.999i$
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  + (−0.597 + 1.28i)2-s + (1.25 + 1.19i)3-s + (−1.28 − 1.53i)4-s + (−0.632 − 1.86i)5-s + (−2.27 + 0.899i)6-s + (−1.03 − 1.35i)7-s + (2.73 − 0.732i)8-s + (0.162 + 2.99i)9-s + (2.76 + 0.302i)10-s + (−5.45 + 0.357i)11-s + (0.208 − 3.45i)12-s + (−4.17 − 4.76i)13-s + (2.35 − 0.522i)14-s + (1.42 − 3.09i)15-s + (−0.694 + 3.93i)16-s + (−2.66 + 2.66i)17-s + ⋯
L(s)  = 1  + (−0.422 + 0.906i)2-s + (0.725 + 0.687i)3-s + (−0.642 − 0.766i)4-s + (−0.282 − 0.832i)5-s + (−0.930 + 0.367i)6-s + (−0.392 − 0.511i)7-s + (0.965 − 0.258i)8-s + (0.0541 + 0.998i)9-s + (0.874 + 0.0957i)10-s + (−1.64 + 0.107i)11-s + (0.0601 − 0.998i)12-s + (−1.15 − 1.32i)13-s + (0.629 − 0.139i)14-s + (0.367 − 0.798i)15-s + (−0.173 + 0.984i)16-s + (−0.646 + 0.646i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0220 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0220 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.0220 + 0.999i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.0220 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.225482 - 0.230509i\)
\(L(\frac12)\) \(\approx\) \(0.225482 - 0.230509i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.597 - 1.28i)T \)
3 \( 1 + (-1.25 - 1.19i)T \)
good5 \( 1 + (0.632 + 1.86i)T + (-3.96 + 3.04i)T^{2} \)
7 \( 1 + (1.03 + 1.35i)T + (-1.81 + 6.76i)T^{2} \)
11 \( 1 + (5.45 - 0.357i)T + (10.9 - 1.43i)T^{2} \)
13 \( 1 + (4.17 + 4.76i)T + (-1.69 + 12.8i)T^{2} \)
17 \( 1 + (2.66 - 2.66i)T - 17iT^{2} \)
19 \( 1 + (0.0844 + 0.424i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (1.98 + 1.52i)T + (5.95 + 22.2i)T^{2} \)
29 \( 1 + (3.02 + 6.14i)T + (-17.6 + 23.0i)T^{2} \)
31 \( 1 + (4.60 - 2.65i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.0591 + 0.297i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (-6.21 - 4.76i)T + (10.6 + 39.6i)T^{2} \)
43 \( 1 + (0.447 + 6.82i)T + (-42.6 + 5.61i)T^{2} \)
47 \( 1 + (-2.77 - 10.3i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.91 - 5.85i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (-3.86 - 11.3i)T + (-46.8 + 35.9i)T^{2} \)
61 \( 1 + (-7.79 + 3.84i)T + (37.1 - 48.3i)T^{2} \)
67 \( 1 + (-0.315 + 4.81i)T + (-66.4 - 8.74i)T^{2} \)
71 \( 1 + (-3.28 + 7.93i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (3.23 + 7.80i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (5.51 - 1.47i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-6.39 - 2.17i)T + (65.8 + 50.5i)T^{2} \)
89 \( 1 + (13.2 + 5.50i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (6.93 + 4.00i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.31222242308352830253950632918, −9.540865785339542695903225408420, −8.581890782862967203697965813152, −7.88175659639862810727750540357, −7.37633475006050877979598679270, −5.71913344142794327993743871060, −4.94383403837310268497775176542, −4.12063254217972523466767468365, −2.53410716119034966965719578427, −0.17632113458736526861261546309, 2.21789281493443060076473103642, 2.68686054549948010413380747270, 3.83787184568706955099519430078, 5.28249977163590102821557728979, 7.00491094694608838576887182407, 7.36130085915483746965862837277, 8.427911768433170299865847907646, 9.291453632597560225989006178210, 9.948402603805335335156948083885, 11.04810765974148811488347698149

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