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Properties

Label 2-24e2-72.61-c1-0-10
Degree $2$
Conductor $576$
Sign $0.406 + 0.913i$
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  + (−1.10 − 1.33i)3-s + (1.5 − 0.866i)5-s + (−0.495 + 0.857i)7-s + (−0.571 + 2.94i)9-s + (1.81 + 1.05i)11-s + (5.50 − 3.18i)13-s + (−2.81 − 1.05i)15-s + 3.81·17-s − 2i·19-s + (1.69 − 0.283i)21-s + (−3.55 − 6.15i)23-s + (−1 + 1.73i)25-s + (4.56 − 2.48i)27-s + (−7.22 − 4.17i)29-s + (1.07 + 1.86i)31-s + ⋯
L(s)  = 1  + (−0.636 − 0.771i)3-s + (0.670 − 0.387i)5-s + (−0.187 + 0.324i)7-s + (−0.190 + 0.981i)9-s + (0.548 + 0.316i)11-s + (1.52 − 0.882i)13-s + (−0.725 − 0.271i)15-s + 0.925·17-s − 0.458i·19-s + (0.369 − 0.0618i)21-s + (−0.740 − 1.28i)23-s + (−0.200 + 0.346i)25-s + (0.878 − 0.477i)27-s + (−1.34 − 0.774i)29-s + (0.193 + 0.335i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.406 + 0.913i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.406 + 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15527 - 0.750124i\)
\(L(\frac12)\) \(\approx\) \(1.15527 - 0.750124i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.10 + 1.33i)T \)
good5 \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.495 - 0.857i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.81 - 1.05i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-5.50 + 3.18i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.81T + 17T^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 + (3.55 + 6.15i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.22 + 4.17i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.07 - 1.86i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.62iT - 37T^{2} \)
41 \( 1 + (0.408 + 0.707i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.97 - 1.14i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.39 + 5.87i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 3.14iT - 53T^{2} \)
59 \( 1 + (-10.3 + 5.95i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.22 - 2.43i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.8 - 6.86i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
79 \( 1 + (4.54 - 7.86i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.71 + 2.14i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 14.0T + 89T^{2} \)
97 \( 1 + (6.22 - 10.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.65733709751245323939624957979, −9.741694065658469439405630507234, −8.716589937747000696634965028372, −7.916307846077139076028138097492, −6.78503731040271571084332081286, −5.87169037914535414070786066289, −5.42766503908488976515522463299, −3.87826547386269729459395364336, −2.26523516830889637338444696514, −0.994793888163488085960880372892, 1.45758119361943598645389012692, 3.45770374209748337553859942620, 4.08095468710394715863010784635, 5.65431533006201122644568932945, 6.06016119309308878229949645122, 7.05774740368524073727597036414, 8.435349835410760782934356064811, 9.445065940705446448983021565681, 9.953570182062532397608783722702, 10.91565479239287797883808901546

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