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Properties

Label 2-546-91.16-c1-0-5
Degree $2$
Conductor $546$
Sign $0.614 - 0.788i$
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 + (−0.5 + 0.866i)3-s + 4-s + (−1.15 + 1.99i)5-s + (0.5 − 0.866i)6-s + (2.61 − 0.396i)7-s − 8-s + (−0.499 − 0.866i)9-s + (1.15 − 1.99i)10-s + (2.92 − 5.06i)11-s + (−0.5 + 0.866i)12-s + (3.58 + 0.349i)13-s + (−2.61 + 0.396i)14-s + (−1.15 − 1.99i)15-s + 16-s − 1.24·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.288 + 0.499i)3-s + 0.5·4-s + (−0.514 + 0.891i)5-s + (0.204 − 0.353i)6-s + (0.988 − 0.149i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.364 − 0.630i)10-s + (0.882 − 1.52i)11-s + (−0.144 + 0.249i)12-s + (0.995 + 0.0968i)13-s + (−0.699 + 0.105i)14-s + (−0.297 − 0.514i)15-s + 0.250·16-s − 0.302·17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.614 - 0.788i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.614 - 0.788i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.928892 + 0.453644i\)
\(L(\frac12)\) \(\approx\) \(0.928892 + 0.453644i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-2.61 + 0.396i)T \)
13 \( 1 + (-3.58 - 0.349i)T \)
good5 \( 1 + (1.15 - 1.99i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.92 + 5.06i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 1.24T + 17T^{2} \)
19 \( 1 + (-2.09 - 3.63i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.94T + 23T^{2} \)
29 \( 1 + (-2.12 - 3.67i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.62 - 4.54i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.04T + 37T^{2} \)
41 \( 1 + (-1.45 - 2.52i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.38 + 9.32i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.13 - 3.69i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.05 - 7.03i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 7.16T + 59T^{2} \)
61 \( 1 + (1.35 + 2.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.66 - 13.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.70 - 8.14i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.25 + 7.37i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.43 + 12.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.24T + 83T^{2} \)
89 \( 1 - 4.56T + 89T^{2} \)
97 \( 1 + (1.71 - 2.97i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.78434031500185330644728075960, −10.42084584099693097453587691314, −8.914399933160987228326020309143, −8.504404192213593136848937324087, −7.42494416909602450632054580172, −6.44684189205141247967299217750, −5.56723419035552377298218098160, −4.00225067383248103795976212340, −3.20101215493312277483138302008, −1.25637239410538920003687187676, 1.00485165213626721672829702043, 2.09395694129389347200796503424, 4.15000580273923653316177954065, 4.99235718646829318851779885301, 6.29964171898456479933590047394, 7.26959323081777700037883493030, 8.083974404480371179426849812802, 8.774594994104581286923102212771, 9.622490156068144063911346561165, 10.78059645377157949046526517035

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