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Properties

Label 2-24e2-144.59-c1-0-3
Degree $2$
Conductor $576$
Sign $-0.628 - 0.778i$
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  + (0.256 + 1.71i)3-s + (0.546 + 2.03i)5-s + (0.0638 + 0.110i)7-s + (−2.86 + 0.878i)9-s + (0.181 − 0.678i)11-s + (0.493 + 1.84i)13-s + (−3.35 + 1.45i)15-s + 4.32i·17-s + (3.97 + 3.97i)19-s + (−0.172 + 0.137i)21-s + (−6.81 − 3.93i)23-s + (0.475 − 0.274i)25-s + (−2.23 − 4.68i)27-s + (−0.248 + 0.926i)29-s + (−4.91 − 2.83i)31-s + ⋯
L(s)  = 1  + (0.148 + 0.988i)3-s + (0.244 + 0.911i)5-s + (0.0241 + 0.0417i)7-s + (−0.956 + 0.292i)9-s + (0.0548 − 0.204i)11-s + (0.136 + 0.511i)13-s + (−0.865 + 0.376i)15-s + 1.04i·17-s + (0.912 + 0.912i)19-s + (−0.0377 + 0.0300i)21-s + (−1.42 − 0.820i)23-s + (0.0951 − 0.0549i)25-s + (−0.431 − 0.902i)27-s + (−0.0461 + 0.172i)29-s + (−0.882 − 0.509i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.628 - 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.628 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.628 - 0.778i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.628 - 0.778i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.603362 + 1.26222i\)
\(L(\frac12)\) \(\approx\) \(0.603362 + 1.26222i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.256 - 1.71i)T \)
good5 \( 1 + (-0.546 - 2.03i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-0.0638 - 0.110i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.181 + 0.678i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.493 - 1.84i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 - 4.32iT - 17T^{2} \)
19 \( 1 + (-3.97 - 3.97i)T + 19iT^{2} \)
23 \( 1 + (6.81 + 3.93i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.248 - 0.926i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (4.91 + 2.83i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.64 + 6.64i)T + 37iT^{2} \)
41 \( 1 + (4.61 - 7.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-10.8 - 2.91i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-5.92 - 10.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.00259 - 0.00259i)T - 53iT^{2} \)
59 \( 1 + (-4.09 + 1.09i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-4.44 - 1.19i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (2.01 - 0.538i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 3.80iT - 71T^{2} \)
73 \( 1 + 1.87iT - 73T^{2} \)
79 \( 1 + (-3.00 + 1.73i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.47 - 0.394i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 10.5T + 89T^{2} \)
97 \( 1 + (-3.31 - 5.74i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.76079760898239954879366296084, −10.28946250034314320573081669810, −9.435547492385783305190793882612, −8.513513243137379691218247514360, −7.57583301304150175840840778206, −6.28605745598711886652393439370, −5.64285425798134821859581581405, −4.22852283767677114377651734654, −3.45341887366654547177300865636, −2.18966266656661956781387098305, 0.78573467951869605145796759347, 2.12032854348750680467225306996, 3.48437542781548202440443261967, 5.07742674051647526618718681326, 5.69805145898364977652437036998, 7.02864599761469126595505202898, 7.61270388299771073795958959522, 8.737636734345034363899493590536, 9.238281238556484377525093218989, 10.37862500611624043034478716733

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