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Properties

Label 2-1584-33.8-c1-0-3
Degree $2$
Conductor $1584$
Sign $-0.930 - 0.365i$
Analytic cond. $12.6483$
Root an. cond. $3.55644$
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  + (−2.23 + 3.07i)5-s + (0.349 + 0.113i)7-s + (2.97 + 1.46i)11-s + (−0.557 − 0.767i)13-s + (2.77 + 2.01i)17-s + (−4.05 + 1.31i)19-s + 4.96i·23-s + (−2.90 − 8.94i)25-s + (−0.767 + 2.36i)29-s + (2.84 − 2.06i)31-s + (−1.12 + 0.820i)35-s + (−2.21 + 6.83i)37-s + (−0.840 − 2.58i)41-s − 1.88i·43-s + (−0.0195 + 0.00636i)47-s + ⋯
L(s)  = 1  + (−0.997 + 1.37i)5-s + (0.132 + 0.0429i)7-s + (0.897 + 0.440i)11-s + (−0.154 − 0.212i)13-s + (0.673 + 0.489i)17-s + (−0.929 + 0.302i)19-s + 1.03i·23-s + (−0.581 − 1.78i)25-s + (−0.142 + 0.438i)29-s + (0.510 − 0.370i)31-s + (−0.190 + 0.138i)35-s + (−0.364 + 1.12i)37-s + (−0.131 − 0.403i)41-s − 0.287i·43-s + (−0.00285 + 0.000928i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 - 0.365i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.930 - 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1584\)    =    \(2^{4} \cdot 3^{2} \cdot 11\)
Sign: $-0.930 - 0.365i$
Analytic conductor: \(12.6483\)
Root analytic conductor: \(3.55644\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1584} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1584,\ (\ :1/2),\ -0.930 - 0.365i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8748748393\)
\(L(\frac12)\) \(\approx\) \(0.8748748393\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 + (-2.97 - 1.46i)T \)
good5 \( 1 + (2.23 - 3.07i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.349 - 0.113i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (0.557 + 0.767i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.77 - 2.01i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (4.05 - 1.31i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 4.96iT - 23T^{2} \)
29 \( 1 + (0.767 - 2.36i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.84 + 2.06i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.21 - 6.83i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.840 + 2.58i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 1.88iT - 43T^{2} \)
47 \( 1 + (0.0195 - 0.00636i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.25 + 4.48i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (6.29 + 2.04i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (5.73 - 7.88i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 4.46T + 67T^{2} \)
71 \( 1 + (6.06 - 8.35i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.18 + 1.35i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (6.42 + 8.84i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (7.42 + 5.39i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 3.04iT - 89T^{2} \)
97 \( 1 + (-12.2 + 8.86i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.03281699411607521238326306863, −8.886550088476241812649360100325, −8.010613041333032875479062059743, −7.37012183836559341708440586683, −6.66717847671921746969236405347, −5.91046096265141088135157024888, −4.58126742112005814913301392013, −3.71660248940786038308417506591, −3.05340813240698837590500620276, −1.69279401703724002701278497399, 0.35379147676267765044238655773, 1.49928189067098936968398429232, 3.10777889546662267074955222092, 4.28278136786035970923910492781, 4.59249321428644806455080675297, 5.72385959817332687078964333069, 6.68170357303705193798599799540, 7.67312180480061487035191319666, 8.349459392637077194664202220907, 8.954530023226268974318557310571

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