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Properties

Label 2-24e2-64.37-c1-0-10
Degree $2$
Conductor $576$
Sign $-0.844 - 0.535i$
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  + (0.809 + 1.15i)2-s + (−0.689 + 1.87i)4-s + (0.159 − 0.799i)5-s + (−0.742 + 1.79i)7-s + (−2.73 + 0.719i)8-s + (1.05 − 0.462i)10-s + (−1.29 + 0.863i)11-s + (1.07 + 5.38i)13-s + (−2.67 + 0.589i)14-s + (−3.04 − 2.58i)16-s + (−1.43 + 1.43i)17-s + (0.0883 − 0.0175i)19-s + (1.39 + 0.849i)20-s + (−2.04 − 0.799i)22-s + (−3.31 + 1.37i)23-s + ⋯
L(s)  = 1  + (0.572 + 0.820i)2-s + (−0.344 + 0.938i)4-s + (0.0711 − 0.357i)5-s + (−0.280 + 0.677i)7-s + (−0.967 + 0.254i)8-s + (0.333 − 0.146i)10-s + (−0.389 + 0.260i)11-s + (0.297 + 1.49i)13-s + (−0.715 + 0.157i)14-s + (−0.762 − 0.647i)16-s + (−0.347 + 0.347i)17-s + (0.0202 − 0.00403i)19-s + (0.311 + 0.190i)20-s + (−0.436 − 0.170i)22-s + (−0.691 + 0.286i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.844 - 0.535i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.844 - 0.535i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.424122 + 1.45974i\)
\(L(\frac12)\) \(\approx\) \(0.424122 + 1.45974i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 1.15i)T \)
3 \( 1 \)
good5 \( 1 + (-0.159 + 0.799i)T + (-4.61 - 1.91i)T^{2} \)
7 \( 1 + (0.742 - 1.79i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (1.29 - 0.863i)T + (4.20 - 10.1i)T^{2} \)
13 \( 1 + (-1.07 - 5.38i)T + (-12.0 + 4.97i)T^{2} \)
17 \( 1 + (1.43 - 1.43i)T - 17iT^{2} \)
19 \( 1 + (-0.0883 + 0.0175i)T + (17.5 - 7.27i)T^{2} \)
23 \( 1 + (3.31 - 1.37i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (-1.04 - 0.695i)T + (11.0 + 26.7i)T^{2} \)
31 \( 1 - 2.58iT - 31T^{2} \)
37 \( 1 + (-1.60 - 0.319i)T + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (0.605 - 0.250i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (5.04 + 7.54i)T + (-16.4 + 39.7i)T^{2} \)
47 \( 1 + (-3.86 + 3.86i)T - 47iT^{2} \)
53 \( 1 + (-8.70 + 5.81i)T + (20.2 - 48.9i)T^{2} \)
59 \( 1 + (1.17 - 5.88i)T + (-54.5 - 22.5i)T^{2} \)
61 \( 1 + (-3.52 + 5.28i)T + (-23.3 - 56.3i)T^{2} \)
67 \( 1 + (-3.15 + 4.72i)T + (-25.6 - 61.8i)T^{2} \)
71 \( 1 + (5.01 - 12.1i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (1.75 + 4.23i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-7.46 - 7.46i)T + 79iT^{2} \)
83 \( 1 + (-2.65 + 0.527i)T + (76.6 - 31.7i)T^{2} \)
89 \( 1 + (7.09 + 2.93i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + 11.4iT - 97T^{2} \)
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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

−11.38160117506371409597951001901, −10.02384191911800038436235084646, −8.920739280607698403426668624370, −8.595233929223322181848466632774, −7.29486644390756807603370610200, −6.53087519723950603467015780269, −5.59604264255977349639235647183, −4.68132962749463833344754650842, −3.66268284763076618967261298391, −2.20931655532548579563606781386, 0.71237231489694412120104824849, 2.55539266822908912546403011354, 3.43274791219497361660505737296, 4.54767476532245249352458108199, 5.64403908077243746857764946696, 6.49732481018810188899176569974, 7.69323471070018539211299614858, 8.744712569455762193828345299490, 9.947272022691784865588902179879, 10.45166437026139161148111829917

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