Properties

Label 2-24e2-576.517-c1-0-77
Degree $2$
Conductor $576$
Sign $0.998 - 0.0453i$
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  + (1.38 − 0.298i)2-s + (0.914 + 1.47i)3-s + (1.82 − 0.826i)4-s + (1.56 − 1.78i)5-s + (1.70 + 1.75i)6-s + (−0.280 + 0.0369i)7-s + (2.27 − 1.68i)8-s + (−1.32 + 2.69i)9-s + (1.62 − 2.93i)10-s + (−0.394 + 0.800i)11-s + (2.88 + 1.92i)12-s + (−0.474 + 1.39i)13-s + (−0.376 + 0.134i)14-s + (4.05 + 0.669i)15-s + (2.63 − 3.01i)16-s + (−3.08 + 3.08i)17-s + ⋯
L(s)  = 1  + (0.977 − 0.211i)2-s + (0.528 + 0.849i)3-s + (0.910 − 0.413i)4-s + (0.699 − 0.797i)5-s + (0.695 + 0.718i)6-s + (−0.106 + 0.0139i)7-s + (0.802 − 0.596i)8-s + (−0.442 + 0.896i)9-s + (0.514 − 0.926i)10-s + (−0.119 + 0.241i)11-s + (0.831 + 0.555i)12-s + (−0.131 + 0.387i)13-s + (−0.100 + 0.0360i)14-s + (1.04 + 0.172i)15-s + (0.658 − 0.752i)16-s + (−0.748 + 0.748i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.998 - 0.0453i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.998 - 0.0453i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.31872 + 0.0752119i\)
\(L(\frac12)\) \(\approx\) \(3.31872 + 0.0752119i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 + 0.298i)T \)
3 \( 1 + (-0.914 - 1.47i)T \)
good5 \( 1 + (-1.56 + 1.78i)T + (-0.652 - 4.95i)T^{2} \)
7 \( 1 + (0.280 - 0.0369i)T + (6.76 - 1.81i)T^{2} \)
11 \( 1 + (0.394 - 0.800i)T + (-6.69 - 8.72i)T^{2} \)
13 \( 1 + (0.474 - 1.39i)T + (-10.3 - 7.91i)T^{2} \)
17 \( 1 + (3.08 - 3.08i)T - 17iT^{2} \)
19 \( 1 + (0.658 + 3.31i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (-1.11 + 8.45i)T + (-22.2 - 5.95i)T^{2} \)
29 \( 1 + (5.19 + 0.340i)T + (28.7 + 3.78i)T^{2} \)
31 \( 1 + (-2.39 - 1.38i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.25 - 6.29i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (0.793 - 6.02i)T + (-39.6 - 10.6i)T^{2} \)
43 \( 1 + (-1.25 - 0.617i)T + (26.1 + 34.1i)T^{2} \)
47 \( 1 + (-4.62 - 1.23i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.782 + 1.17i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (-7.46 + 8.50i)T + (-7.70 - 58.4i)T^{2} \)
61 \( 1 + (0.229 - 3.50i)T + (-60.4 - 7.96i)T^{2} \)
67 \( 1 + (11.6 - 5.76i)T + (40.7 - 53.1i)T^{2} \)
71 \( 1 + (0.728 - 1.75i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (4.44 + 10.7i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (0.222 - 0.828i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (2.40 - 2.11i)T + (10.8 - 82.2i)T^{2} \)
89 \( 1 + (7.90 + 3.27i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (4.93 - 2.84i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.71678801198061841089007014940, −9.952233341618167499388360025121, −9.112239900607582253571365129526, −8.285232400251113642239998331366, −6.87497436112144616382850756503, −5.88221027805192937408462623049, −4.78546809307988190146262266678, −4.37169454324469124032465237094, −2.94881281984976397996319744002, −1.88446398022282132303005958960, 1.91070391812751564746272640827, 2.84183600000042167060719466675, 3.79449146182895744970624391264, 5.47483984270758667344077175109, 6.11625679974360117337548847712, 7.11688793126186810159262943182, 7.60007432922551928216126486611, 8.825156125354236989164776486130, 9.910204327707839634076920665822, 10.96386456144056487143386703093

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