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Properties

Label 2-1512-1512.1021-c0-0-1
Degree $2$
Conductor $1512$
Sign $0.448 - 0.893i$
Analytic cond. $0.754586$
Root an. cond. $0.868669$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.766 − 0.642i)2-s + (0.5 + 0.866i)3-s + (0.173 − 0.984i)4-s + (−1.76 + 0.642i)5-s + (0.939 + 0.342i)6-s + (0.173 + 0.984i)7-s + (−0.500 − 0.866i)8-s + (−0.499 + 0.866i)9-s + (−0.939 + 1.62i)10-s + (0.939 − 0.342i)12-s + (0.766 + 0.642i)13-s + (0.766 + 0.642i)14-s + (−1.43 − 1.20i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)18-s + (0.173 + 0.300i)19-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.5 + 0.866i)3-s + (0.173 − 0.984i)4-s + (−1.76 + 0.642i)5-s + (0.939 + 0.342i)6-s + (0.173 + 0.984i)7-s + (−0.500 − 0.866i)8-s + (−0.499 + 0.866i)9-s + (−0.939 + 1.62i)10-s + (0.939 − 0.342i)12-s + (0.766 + 0.642i)13-s + (0.766 + 0.642i)14-s + (−1.43 − 1.20i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)18-s + (0.173 + 0.300i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.448 - 0.893i$
Analytic conductor: \(0.754586\)
Root analytic conductor: \(0.868669\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (1021, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :0),\ 0.448 - 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.349557845\)
\(L(\frac12)\) \(\approx\) \(1.349557845\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 + 0.642i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.173 - 0.984i)T \)
good5 \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \)
11 \( 1 + (-0.766 - 0.642i)T^{2} \)
13 \( 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.939 + 0.342i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.173 - 0.984i)T^{2} \)
43 \( 1 + (-0.766 - 0.642i)T^{2} \)
47 \( 1 + (0.939 - 0.342i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-1.87 + 0.684i)T + (0.766 - 0.642i)T^{2} \)
61 \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (-0.173 - 0.984i)T^{2} \)
71 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
83 \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.766 - 0.642i)T^{2} \)
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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

−9.886444321101022172110579428437, −9.114952655861471386966700665097, −8.295381787836537238964274245200, −7.52678742585252470890586612506, −6.40596660251561964991236748421, −5.37672425588564750765151350971, −4.51622457574136755048623136667, −3.59287874635185335457275662004, −3.32791386716660817160323228668, −2.05265841945606898454214609829, 0.826082195581700843991105819901, 2.79358374711878727284996130663, 3.81824972038186660433067743128, 4.22235370145455635383042878382, 5.32630609007917896063392875805, 6.59450438517177953073031604181, 7.13738748432169065925737663307, 7.957126945048532260898093504836, 8.267377806203882910330444074930, 8.930169819553289886574223132285

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