Properties

Label 2-546-273.257-c1-0-0
Degree $2$
Conductor $546$
Sign $-0.996 + 0.0880i$
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  − 2-s + (1.21 + 1.23i)3-s + 4-s + (−1.57 − 0.908i)5-s + (−1.21 − 1.23i)6-s + (−2.47 + 0.947i)7-s − 8-s + (−0.0353 + 2.99i)9-s + (1.57 + 0.908i)10-s + (1.02 − 1.77i)11-s + (1.21 + 1.23i)12-s + (−3.57 + 0.463i)13-s + (2.47 − 0.947i)14-s + (−0.796 − 3.04i)15-s + 16-s − 5.68·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.702 + 0.711i)3-s + 0.5·4-s + (−0.703 − 0.406i)5-s + (−0.497 − 0.502i)6-s + (−0.933 + 0.358i)7-s − 0.353·8-s + (−0.0117 + 0.999i)9-s + (0.497 + 0.287i)10-s + (0.309 − 0.535i)11-s + (0.351 + 0.355i)12-s + (−0.991 + 0.128i)13-s + (0.660 − 0.253i)14-s + (−0.205 − 0.786i)15-s + 0.250·16-s − 1.37·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.996 + 0.0880i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.996 + 0.0880i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0103116 - 0.233747i\)
\(L(\frac12)\) \(\approx\) \(0.0103116 - 0.233747i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-1.21 - 1.23i)T \)
7 \( 1 + (2.47 - 0.947i)T \)
13 \( 1 + (3.57 - 0.463i)T \)
good5 \( 1 + (1.57 + 0.908i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.02 + 1.77i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 5.68T + 17T^{2} \)
19 \( 1 + (0.796 + 1.38i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 8.73iT - 23T^{2} \)
29 \( 1 + (0.724 - 0.418i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.97 + 6.88i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.21iT - 37T^{2} \)
41 \( 1 + (0.397 - 0.229i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.836 + 1.44i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.94 - 1.12i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.497 - 0.286i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 2.19iT - 59T^{2} \)
61 \( 1 + (5.97 - 3.44i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.46 + 2.57i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.24 - 7.34i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.11 - 8.86i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.68 - 6.38i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.0iT - 83T^{2} \)
89 \( 1 - 3.86iT - 89T^{2} \)
97 \( 1 + (1.72 - 2.98i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.19568772127196106213665919955, −10.07624512377497149999251345695, −9.319727327121057896112665188508, −8.871886775437934056967210666487, −7.88721163528356953446657044868, −7.04739379527837260151088789806, −5.76291354248580493132247750053, −4.44435602782295764095082941302, −3.44121177141764925613397033315, −2.28691559884651409369582360593, 0.14253174107031891466824402737, 2.12867149833423758270933282953, 3.18268876784789527685414788490, 4.33860644644166896532665189849, 6.31134258976212968605269413496, 7.02089776102078591559919591636, 7.50746191829973409822572359082, 8.631315568246097335149303516332, 9.298752010650306570330562241839, 10.24259560613899373098634897573

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