Properties

Label 2-546-91.9-c1-0-0
Degree $2$
Conductor $546$
Sign $-0.994 + 0.107i$
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 − 3-s + (−0.499 + 0.866i)4-s + (0.623 − 1.07i)5-s + (−0.5 − 0.866i)6-s + (−2.27 + 1.34i)7-s − 0.999·8-s + 9-s + 1.24·10-s − 2.49·11-s + (0.499 − 0.866i)12-s + (−0.785 + 3.51i)13-s + (−2.30 − 1.30i)14-s + (−0.623 + 1.07i)15-s + (−0.5 − 0.866i)16-s + (−0.247 + 0.428i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s − 0.577·3-s + (−0.249 + 0.433i)4-s + (0.278 − 0.482i)5-s + (−0.204 − 0.353i)6-s + (−0.861 + 0.507i)7-s − 0.353·8-s + 0.333·9-s + 0.394·10-s − 0.752·11-s + (0.144 − 0.249i)12-s + (−0.217 + 0.976i)13-s + (−0.615 − 0.348i)14-s + (−0.160 + 0.278i)15-s + (−0.125 − 0.216i)16-s + (−0.0600 + 0.104i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.107i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.994 + 0.107i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.994 + 0.107i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0257968 - 0.476417i\)
\(L(\frac12)\) \(\approx\) \(0.0257968 - 0.476417i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + (2.27 - 1.34i)T \)
13 \( 1 + (0.785 - 3.51i)T \)
good5 \( 1 + (-0.623 + 1.07i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + 2.49T + 11T^{2} \)
17 \( 1 + (0.247 - 0.428i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 7.67T + 19T^{2} \)
23 \( 1 + (0.224 + 0.388i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.71 - 6.42i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.06 + 7.03i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.37 + 2.37i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.87 - 3.24i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.47 - 2.55i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.29 + 5.71i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.86 - 6.69i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.727 + 1.26i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 4.19T + 61T^{2} \)
67 \( 1 - 0.0276T + 67T^{2} \)
71 \( 1 + (-4.68 - 8.12i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.07 + 8.78i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.93 - 8.54i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.42T + 83T^{2} \)
89 \( 1 + (-5.74 - 9.94i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.509 + 0.883i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.31234882714553528367836798478, −10.37890001134887257441892537741, −9.281591487409931205493001606064, −8.738457051728435427201872528083, −7.43699028913705864130689625417, −6.54213684573096092886604001218, −5.78114836600990286504409952021, −4.92063839635828558199678021956, −3.83606540649433245123022017920, −2.22507047565512331569853501570, 0.24349707769701556348661131074, 2.30036888111904558541440380672, 3.41859210599648905175481375319, 4.58061126985932387520000782322, 5.71493359495347451579022329100, 6.47323740074298347061270520110, 7.46900278405473406711140545850, 8.732667877719737847906255102475, 9.984922913747272696423466246436, 10.45073831686653567235133032318

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