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Properties

Label 2-546-91.4-c1-0-2
Degree $2$
Conductor $546$
Sign $-0.750 - 0.660i$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
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  − i·2-s + (0.5 + 0.866i)3-s − 4-s + (−0.825 + 0.476i)5-s + (0.866 − 0.5i)6-s + (−2.63 − 0.261i)7-s + i·8-s + (−0.499 + 0.866i)9-s + (0.476 + 0.825i)10-s + (0.0637 − 0.0368i)11-s + (−0.5 − 0.866i)12-s + (−3.60 − 0.173i)13-s + (−0.261 + 2.63i)14-s + (−0.825 − 0.476i)15-s + 16-s − 2.20·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.288 + 0.499i)3-s − 0.5·4-s + (−0.368 + 0.213i)5-s + (0.353 − 0.204i)6-s + (−0.995 − 0.0988i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.150 + 0.260i)10-s + (0.0192 − 0.0111i)11-s + (−0.144 − 0.249i)12-s + (−0.998 − 0.0482i)13-s + (−0.0699 + 0.703i)14-s + (−0.213 − 0.122i)15-s + 0.250·16-s − 0.533·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.750 - 0.660i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.750 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.750 - 0.660i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.750 - 0.660i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0794756 + 0.210672i\)
\(L(\frac12)\) \(\approx\) \(0.0794756 + 0.210672i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (2.63 + 0.261i)T \)
13 \( 1 + (3.60 + 0.173i)T \)
good5 \( 1 + (0.825 - 0.476i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.0637 + 0.0368i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 2.20T + 17T^{2} \)
19 \( 1 + (0.747 + 0.431i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 8.03T + 23T^{2} \)
29 \( 1 + (2.36 - 4.10i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-8.72 - 5.03i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 11.2iT - 37T^{2} \)
41 \( 1 + (5.80 + 3.35i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.23 + 2.14i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (8.12 - 4.68i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.935 + 1.62i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 13.5iT - 59T^{2} \)
61 \( 1 + (1.78 - 3.09i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.5 + 6.10i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8.96 + 5.17i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.76 - 3.90i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.30 + 2.25i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.8iT - 83T^{2} \)
89 \( 1 + 3.63iT - 89T^{2} \)
97 \( 1 + (-11.1 + 6.42i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.00628967508290582453529460396, −10.18503452029446550998005267101, −9.616303413569476798837291647268, −8.746723994996975627386580501603, −7.68480816809065429075147220995, −6.64820655547465754263118823792, −5.37047045885252728922773646334, −4.19148643546670149084309481766, −3.37461463164794287879031105378, −2.27118124369825206277496307852, 0.11655987403281761737740896794, 2.37287121678518858364714997789, 3.76075547219901086840044592740, 4.83627759855881959197999307050, 6.25349144328052054894138606004, 6.65314417070329427938932526191, 7.953812850188579645455308175808, 8.286055014791132650776040663721, 9.737277331314684480650270702241, 9.907179673940005908048906012947

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