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Properties

Label 2-24e2-72.61-c1-0-13
Degree $2$
Conductor $576$
Sign $0.913 - 0.406i$
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

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

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.913 - 0.406i$
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: \((2,\ 576,\ (\ :1/2),\ 0.913 - 0.406i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04421 + 0.434668i\)
\(L(\frac12)\) \(\approx\) \(2.04421 + 0.434668i\)
\(L(\frac{3}{2})\) 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

−10.76092421932899486370120935638, −9.773045291374258281115365020308, −9.190800992375141351828556897344, −8.148121839614429220915833028823, −7.58955370801359674037913968051, −5.74692603817308376619633510332, −5.43941642227534681569790531137, −3.94246686561882859391575368183, −3.14443462230379718564202655180, −1.53065952057537397413287902193, 1.49383052984470505889186830968, 2.57053121147095429392307065330, 3.70732181758465814920163702139, 5.26647816365502704010810002135, 6.34989326607331177296339933341, 6.95097159200580855651182602929, 8.126670741123651943448538279472, 8.802984610918814415203359644184, 9.644819067364928997058596662105, 10.68272778987035795896680247349

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