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Properties

Label 2-546-273.101-c1-0-14
Degree $2$
Conductor $546$
Sign $0.788 - 0.614i$
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 + (1.17 + 1.27i)3-s + (−0.499 − 0.866i)4-s + (−0.870 + 0.502i)5-s + (1.69 − 0.378i)6-s + (1.33 + 2.28i)7-s − 0.999·8-s + (−0.249 + 2.98i)9-s + 1.00i·10-s − 0.620·11-s + (0.517 − 1.65i)12-s + (1.14 + 3.41i)13-s + (2.64 − 0.0151i)14-s + (−1.66 − 0.520i)15-s + (−0.5 + 0.866i)16-s + (−0.171 − 0.296i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.677 + 0.735i)3-s + (−0.249 − 0.433i)4-s + (−0.389 + 0.224i)5-s + (0.690 − 0.154i)6-s + (0.504 + 0.863i)7-s − 0.353·8-s + (−0.0831 + 0.996i)9-s + 0.317i·10-s − 0.187·11-s + (0.149 − 0.477i)12-s + (0.316 + 0.948i)13-s + (0.707 − 0.00404i)14-s + (−0.428 − 0.134i)15-s + (−0.125 + 0.216i)16-s + (−0.0415 − 0.0720i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 - 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.788 - 0.614i$
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: \((2,\ 546,\ (\ :1/2),\ 0.788 - 0.614i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89401 + 0.650975i\)
\(L(\frac12)\) \(\approx\) \(1.89401 + 0.650975i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.17 - 1.27i)T \)
7 \( 1 + (-1.33 - 2.28i)T \)
13 \( 1 + (-1.14 - 3.41i)T \)
good5 \( 1 + (0.870 - 0.502i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + 0.620T + 11T^{2} \)
17 \( 1 + (0.171 + 0.296i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 4.33T + 19T^{2} \)
23 \( 1 + (2.44 + 1.41i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-8.23 + 4.75i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.25 + 2.16i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.76 + 2.17i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.47 + 4.31i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.602 - 1.04i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.0442 - 0.0255i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.15 + 2.39i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.67 - 1.54i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 6.68iT - 61T^{2} \)
67 \( 1 - 5.48iT - 67T^{2} \)
71 \( 1 + (-0.621 + 1.07i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.46 + 7.72i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.458 - 0.793i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 13.2iT - 83T^{2} \)
89 \( 1 + (3.11 + 1.80i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.10 + 8.84i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.03402325222234141479718487649, −9.983786978655698803443861684860, −9.250828988268666392244234426969, −8.453309581778489425382950219715, −7.54692606168794017400629813451, −6.04578978237733595602945007133, −4.96785246023830822625623341717, −4.12192274152835201533702972343, −3.04009550952304711425742904686, −1.99361911841802184357133143245, 1.06464861542914784290656508852, 2.94792378473202687478537707259, 3.96203131038036645818737790135, 5.09796510675481920684979507725, 6.31175187534099982912291665444, 7.25828377284416091479987254281, 7.995660145540414181234552588768, 8.428511153141167928943650004663, 9.688681643082578418097393050338, 10.71749889447601782071034193652

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