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Properties

Label 2-24e2-64.29-c1-0-7
Degree $2$
Conductor $576$
Sign $0.873 - 0.487i$
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  + (0.662 − 1.24i)2-s + (−1.12 − 1.65i)4-s + (−1.62 + 2.43i)5-s + (0.294 − 0.121i)7-s + (−2.81 + 0.303i)8-s + (1.96 + 3.64i)10-s + (1.07 + 5.42i)11-s + (2.67 + 4.00i)13-s + (0.0427 − 0.448i)14-s + (−1.48 + 3.71i)16-s + (−0.394 − 0.394i)17-s + (−2.13 + 1.42i)19-s + (5.84 − 0.0360i)20-s + (7.49 + 2.24i)22-s + (3.37 − 8.14i)23-s + ⋯
L(s)  = 1  + (0.468 − 0.883i)2-s + (−0.560 − 0.828i)4-s + (−0.726 + 1.08i)5-s + (0.111 − 0.0460i)7-s + (−0.994 + 0.107i)8-s + (0.620 + 1.15i)10-s + (0.325 + 1.63i)11-s + (0.742 + 1.11i)13-s + (0.0114 − 0.119i)14-s + (−0.371 + 0.928i)16-s + (−0.0956 − 0.0956i)17-s + (−0.490 + 0.327i)19-s + (1.30 − 0.00806i)20-s + (1.59 + 0.479i)22-s + (0.703 − 1.69i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 - 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.873 - 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.873 - 0.487i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.873 - 0.487i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28993 + 0.335814i\)
\(L(\frac12)\) \(\approx\) \(1.28993 + 0.335814i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.662 + 1.24i)T \)
3 \( 1 \)
good5 \( 1 + (1.62 - 2.43i)T + (-1.91 - 4.61i)T^{2} \)
7 \( 1 + (-0.294 + 0.121i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-1.07 - 5.42i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 + (-2.67 - 4.00i)T + (-4.97 + 12.0i)T^{2} \)
17 \( 1 + (0.394 + 0.394i)T + 17iT^{2} \)
19 \( 1 + (2.13 - 1.42i)T + (7.27 - 17.5i)T^{2} \)
23 \( 1 + (-3.37 + 8.14i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (0.411 - 2.06i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 - 1.31iT - 31T^{2} \)
37 \( 1 + (-1.47 - 0.985i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (4.39 - 10.6i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (2.25 - 0.448i)T + (39.7 - 16.4i)T^{2} \)
47 \( 1 + (-5.23 - 5.23i)T + 47iT^{2} \)
53 \( 1 + (1.88 + 9.48i)T + (-48.9 + 20.2i)T^{2} \)
59 \( 1 + (-0.373 + 0.559i)T + (-22.5 - 54.5i)T^{2} \)
61 \( 1 + (-0.868 - 0.172i)T + (56.3 + 23.3i)T^{2} \)
67 \( 1 + (-10.3 - 2.04i)T + (61.8 + 25.6i)T^{2} \)
71 \( 1 + (-12.4 + 5.14i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (14.5 + 6.01i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (2.55 - 2.55i)T - 79iT^{2} \)
83 \( 1 + (0.585 - 0.391i)T + (31.7 - 76.6i)T^{2} \)
89 \( 1 + (3.72 + 8.98i)T + (-62.9 + 62.9i)T^{2} \)
97 \( 1 + 0.565iT - 97T^{2} \)
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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

−10.97227494820307957062130748931, −10.16821592339908267422446595812, −9.308682183420019929054461425154, −8.251018683608283225937509894494, −6.88409696829125654493329281426, −6.47147837544340970043519324831, −4.72922719492216639627730156398, −4.12891881664316599192769903862, −2.97584605724295006340558482832, −1.77188887018271322565125616698, 0.68319829320561400968119592259, 3.30797090776402714512875351775, 4.03035268325648114452084578473, 5.32056200164488112697653431345, 5.80894546727012691967172060994, 7.09066823246839080368289957427, 8.236288764806313900205089621649, 8.466564907866812469124827408439, 9.339574492395492822054430666689, 10.90771466959899984365236473187

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