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Properties

Label 2-546-273.257-c1-0-28
Degree $2$
Conductor $546$
Sign $0.190 + 0.981i$
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 + (1.72 + 0.0900i)3-s + 4-s + (−1.98 − 1.14i)5-s + (−1.72 − 0.0900i)6-s + (−0.877 − 2.49i)7-s − 8-s + (2.98 + 0.311i)9-s + (1.98 + 1.14i)10-s + (−0.148 + 0.257i)11-s + (1.72 + 0.0900i)12-s + (3.20 + 1.66i)13-s + (0.877 + 2.49i)14-s + (−3.32 − 2.15i)15-s + 16-s + 0.893·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.998 + 0.0519i)3-s + 0.5·4-s + (−0.886 − 0.511i)5-s + (−0.706 − 0.0367i)6-s + (−0.331 − 0.943i)7-s − 0.353·8-s + (0.994 + 0.103i)9-s + (0.626 + 0.361i)10-s + (−0.0447 + 0.0775i)11-s + (0.499 + 0.0259i)12-s + (0.887 + 0.460i)13-s + (0.234 + 0.667i)14-s + (−0.858 − 0.557i)15-s + 0.250·16-s + 0.216·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.190 + 0.981i$
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: \((2,\ 546,\ (\ :1/2),\ 0.190 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.887425 - 0.731897i\)
\(L(\frac12)\) \(\approx\) \(0.887425 - 0.731897i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 - 0.0900i)T \)
7 \( 1 + (0.877 + 2.49i)T \)
13 \( 1 + (-3.20 - 1.66i)T \)
good5 \( 1 + (1.98 + 1.14i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.148 - 0.257i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 0.893T + 17T^{2} \)
19 \( 1 + (3.94 + 6.83i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.81iT - 23T^{2} \)
29 \( 1 + (0.980 - 0.566i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.839 + 1.45i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.99iT - 37T^{2} \)
41 \( 1 + (-6.52 + 3.76i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.94 - 3.36i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.21 - 3.00i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.28 - 3.62i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 6.02iT - 59T^{2} \)
61 \( 1 + (-7.31 + 4.22i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.94 - 1.69i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.14 - 1.99i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.16 - 10.6i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.46 + 7.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.54iT - 83T^{2} \)
89 \( 1 - 14.3iT - 89T^{2} \)
97 \( 1 + (1.19 - 2.06i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.61723294110689084502011608479, −9.477928927841597143678745979895, −8.772981533757668136879706113339, −8.119566717810452932349341024310, −7.24874960257806598260597403658, −6.48812568387395564564688144999, −4.48770109498329317875578592369, −3.87310626159576129282521015627, −2.49296726288061717287599045765, −0.77207983010443259117171637343, 1.77526722413401910996639005726, 3.16936262659960466474962763089, 3.78846177162922805676020669514, 5.67986141667151571126137865378, 6.71038867387877187702723882394, 7.86436417011213450633194873251, 8.176568057026644260194670963078, 9.144846135606178789230822953377, 9.936917557767584700625370316037, 10.86118532261291517142750223644

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