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Properties

Label 2-24e2-9.7-c1-0-15
Degree $2$
Conductor $576$
Sign $0.972 + 0.234i$
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  + (1.68 − 0.396i)3-s + (1.18 + 2.05i)5-s + (2.18 − 3.78i)7-s + (2.68 − 1.33i)9-s + (−0.5 + 0.866i)11-s + (−0.186 − 0.322i)13-s + (2.81 + 2.99i)15-s − 5.37·17-s + 0.627·19-s + (2.18 − 7.25i)21-s + (−0.186 − 0.322i)23-s + (−0.313 + 0.543i)25-s + (4 − 3.31i)27-s + (−2.18 + 3.78i)29-s + (3.18 + 5.51i)31-s + ⋯
L(s)  = 1  + (0.973 − 0.228i)3-s + (0.530 + 0.918i)5-s + (0.826 − 1.43i)7-s + (0.895 − 0.445i)9-s + (−0.150 + 0.261i)11-s + (−0.0516 − 0.0894i)13-s + (0.726 + 0.773i)15-s − 1.30·17-s + 0.144·19-s + (0.477 − 1.58i)21-s + (−0.0388 − 0.0672i)23-s + (−0.0627 + 0.108i)25-s + (0.769 − 0.638i)27-s + (−0.405 + 0.703i)29-s + (0.572 + 0.991i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.972 + 0.234i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.972 + 0.234i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.30972 - 0.274545i\)
\(L(\frac12)\) \(\approx\) \(2.30972 - 0.274545i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 + 0.396i)T \)
good5 \( 1 + (-1.18 - 2.05i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.18 + 3.78i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.186 + 0.322i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.37T + 17T^{2} \)
19 \( 1 - 0.627T + 19T^{2} \)
23 \( 1 + (0.186 + 0.322i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.18 - 3.78i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.18 - 5.51i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.74T + 37T^{2} \)
41 \( 1 + (-5.87 - 10.1i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.872 + 1.51i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.18 + 3.78i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.744T + 53T^{2} \)
59 \( 1 + (3.5 + 6.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.18 + 2.05i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.87 - 3.24i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 + (3.18 - 5.51i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.81 - 8.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (0.872 - 1.51i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.55773735787710595146810296296, −9.976292579623248993287314234192, −8.862098348518674798477442440116, −7.951572945331293578745150007017, −7.07559063708724192077585617480, −6.61851754281682846513484341495, −4.86640272009447604577734518458, −3.89823593214760002329915209241, −2.72255292496608919843230168757, −1.54026576270405404934240011293, 1.79180421080128494628330402398, 2.59305142203443513505009289853, 4.22826253247854884092107840023, 5.11289102859174306019458893538, 5.97254022205530949569961212358, 7.46557633662228301988868131620, 8.475403487644185845381834060514, 8.921780912729909629948403034189, 9.480726588525398554762480060873, 10.69000255154661215001915697695

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