Properties

Label 2-24e2-576.205-c1-0-73
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: $\chi_{576} (205, \cdot )$
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

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

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