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Properties

Label 2-24e2-12.11-c1-0-2
Degree $2$
Conductor $576$
Sign $0.577 - 0.816i$
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  + 1.41i·5-s + 4·13-s + 7.07i·17-s + 2.99·25-s + 9.89i·29-s − 2·37-s − 1.41i·41-s + 7·49-s − 7.07i·53-s + 10·61-s + 5.65i·65-s − 16·73-s − 10.0·85-s − 18.3i·89-s + 8·97-s + ⋯
L(s)  = 1  + 0.632i·5-s + 1.10·13-s + 1.71i·17-s + 0.599·25-s + 1.83i·29-s − 0.328·37-s − 0.220i·41-s + 49-s − 0.971i·53-s + 1.28·61-s + 0.701i·65-s − 1.87·73-s − 1.08·85-s − 1.94i·89-s + 0.812·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.577 - 0.816i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29692 + 0.671335i\)
\(L(\frac12)\) \(\approx\) \(1.29692 + 0.671335i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 - 1.41iT - 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 - 7.07iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 9.89iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 1.41iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 7.07iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 16T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 18.3iT - 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−10.68275052586659087921319497973, −10.31670034381543091886270597365, −8.915288129991793242359600146610, −8.365345937509952469990148640623, −7.15726967605187408569971082822, −6.37892264030131945486383552163, −5.45561806442231171703791859695, −4.04403104942718968977940689424, −3.15271126923595400466103256639, −1.59127865434395909682260622352, 0.927493852611657402345888739097, 2.62048424389790237137731620764, 3.97991842024068186800107769066, 4.98753274533784914708833691558, 5.95231434978599084789235368728, 7.01748190609164655772193904811, 8.022253874225529949051092210267, 8.892310306806202347208647108774, 9.575390039087647071437045154020, 10.61885454202765275179954157864

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