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Properties

Label 2-546-273.101-c1-0-28
Degree $2$
Conductor $546$
Sign $-0.792 + 0.609i$
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  + (0.5 − 0.866i)2-s + (−0.458 + 1.67i)3-s + (−0.499 − 0.866i)4-s + (−1.57 + 0.908i)5-s + (1.21 + 1.23i)6-s + (−0.414 − 2.61i)7-s − 0.999·8-s + (−2.58 − 1.53i)9-s + 1.81i·10-s − 2.05·11-s + (1.67 − 0.438i)12-s + (3.57 − 0.463i)13-s + (−2.47 − 0.947i)14-s + (−0.796 − 3.04i)15-s + (−0.5 + 0.866i)16-s + (−2.84 − 4.92i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.264 + 0.964i)3-s + (−0.249 − 0.433i)4-s + (−0.703 + 0.406i)5-s + (0.497 + 0.502i)6-s + (−0.156 − 0.987i)7-s − 0.353·8-s + (−0.860 − 0.510i)9-s + 0.574i·10-s − 0.618·11-s + (0.483 − 0.126i)12-s + (0.991 − 0.128i)13-s + (−0.660 − 0.253i)14-s + (−0.205 − 0.786i)15-s + (−0.125 + 0.216i)16-s + (−0.689 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.792 + 0.609i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.792 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.792 + 0.609i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.792 + 0.609i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.182229 - 0.535545i\)
\(L(\frac12)\) \(\approx\) \(0.182229 - 0.535545i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.458 - 1.67i)T \)
7 \( 1 + (0.414 + 2.61i)T \)
13 \( 1 + (-3.57 + 0.463i)T \)
good5 \( 1 + (1.57 - 0.908i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + 2.05T + 11T^{2} \)
17 \( 1 + (2.84 + 4.92i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 1.59T + 19T^{2} \)
23 \( 1 + (7.56 + 4.36i)T + (11.5 + 19.9i)T^{2} \)
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 \( 1 + (3.65 + 2.10i)T + (18.5 + 32.0i)T^{2} \)
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 \( 1 + (-1.89 + 1.09i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 6.89iT - 61T^{2} \)
67 \( 1 - 5.15iT - 67T^{2} \)
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 \( 1 + (-3.34 - 1.93i)T + (44.5 + 77.0i)T^{2} \)
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

−10.50147093837818040563297279854, −10.00516897442127077792551246991, −8.868830266612682121929822966005, −7.85508188270457906719053607868, −6.66427692257547296262487513888, −5.60008977312270436927323291591, −4.31442214430103894374344268670, −3.89513627874471509438899250545, −2.70641709676041776654765027236, −0.28860173401644939325655735491, 1.94867279287614935074513784547, 3.48296105893230732843558356346, 4.76720500648655938515858582337, 5.94709232970716066764830295059, 6.34098659407130594987815334645, 7.66463731678553786174001629450, 8.341759340914094805690490171239, 8.823520457471327136896522215124, 10.42113752342901619808447451694, 11.51959494751129962174453870849

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