Properties

Label 2-12e3-216.5-c0-0-0
Degree $2$
Conductor $1728$
Sign $-0.880 + 0.474i$
Analytic cond. $0.862384$
Root an. cond. $0.928646$
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  + (−0.342 + 0.939i)3-s + (−0.766 − 0.642i)9-s + (−1.85 − 0.673i)11-s + (−1.11 − 0.642i)17-s + (−1.62 + 0.939i)19-s + (−0.766 + 0.642i)25-s + (0.866 − 0.500i)27-s + (1.26 − 1.50i)33-s + (−0.439 + 0.524i)41-s + (−0.524 + 1.43i)43-s + (0.939 − 0.342i)49-s + (0.984 − 0.826i)51-s + (−0.326 − 1.85i)57-s + (0.642 − 0.233i)59-s + (0.223 − 0.266i)67-s + ⋯
L(s)  = 1  + (−0.342 + 0.939i)3-s + (−0.766 − 0.642i)9-s + (−1.85 − 0.673i)11-s + (−1.11 − 0.642i)17-s + (−1.62 + 0.939i)19-s + (−0.766 + 0.642i)25-s + (0.866 − 0.500i)27-s + (1.26 − 1.50i)33-s + (−0.439 + 0.524i)41-s + (−0.524 + 1.43i)43-s + (0.939 − 0.342i)49-s + (0.984 − 0.826i)51-s + (−0.326 − 1.85i)57-s + (0.642 − 0.233i)59-s + (0.223 − 0.266i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.880 + 0.474i$
Analytic conductor: \(0.862384\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (545, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :0),\ -0.880 + 0.474i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07547217387\)
\(L(\frac12)\) \(\approx\) \(0.07547217387\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.342 - 0.939i)T \)
good5 \( 1 + (0.766 - 0.642i)T^{2} \)
7 \( 1 + (-0.939 + 0.342i)T^{2} \)
11 \( 1 + (1.85 + 0.673i)T + (0.766 + 0.642i)T^{2} \)
13 \( 1 + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (1.11 + 0.642i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (1.62 - 0.939i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.939 + 0.342i)T^{2} \)
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 \( 1 + (0.439 - 0.524i)T + (-0.173 - 0.984i)T^{2} \)
43 \( 1 + (0.524 - 1.43i)T + (-0.766 - 0.642i)T^{2} \)
47 \( 1 + (0.939 - 0.342i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.642 + 0.233i)T + (0.766 - 0.642i)T^{2} \)
61 \( 1 + (0.939 - 0.342i)T^{2} \)
67 \( 1 + (-0.223 + 0.266i)T + (-0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.173 - 0.984i)T^{2} \)
83 \( 1 + (-1.32 + 1.11i)T + (0.173 - 0.984i)T^{2} \)
89 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \)
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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

−10.12741376111804163669649407044, −9.230105145454465726566553735133, −8.420987206366338855403803610927, −7.83113628179333604054734324894, −6.57448684484332298736788399220, −5.81306530000810734472989382502, −5.05840624802298054429203544970, −4.28173904648849770597874681341, −3.24593735223810818633333838397, −2.27734383371600389738130200546, 0.05297548025337466585771774769, 2.10584285879733726371787476650, 2.50191447059616628307600451980, 4.17936523235121832179649941070, 5.06410063969083840094979171781, 5.87900097477039194291267308570, 6.78105929337533354484647529885, 7.34655867456190940433767926775, 8.330312680599118642305784673137, 8.673443935247184714884106813458

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