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Properties

Label 2-546-273.257-c1-0-9
Degree $2$
Conductor $546$
Sign $-0.197 - 0.980i$
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 + (0.805 + 1.53i)3-s + 4-s + (2.64 + 1.52i)5-s + (−0.805 − 1.53i)6-s + (0.969 + 2.46i)7-s − 8-s + (−1.70 + 2.46i)9-s + (−2.64 − 1.52i)10-s + (−1.13 + 1.96i)11-s + (0.805 + 1.53i)12-s + (3.11 − 1.81i)13-s + (−0.969 − 2.46i)14-s + (−0.212 + 5.29i)15-s + 16-s − 3.59·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.464 + 0.885i)3-s + 0.5·4-s + (1.18 + 0.684i)5-s + (−0.328 − 0.626i)6-s + (0.366 + 0.930i)7-s − 0.353·8-s + (−0.567 + 0.823i)9-s + (−0.837 − 0.483i)10-s + (−0.341 + 0.591i)11-s + (0.232 + 0.442i)12-s + (0.864 − 0.502i)13-s + (−0.259 − 0.657i)14-s + (−0.0548 + 1.36i)15-s + 0.250·16-s − 0.872·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.197 - 0.980i$
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: \((2,\ 546,\ (\ :1/2),\ -0.197 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.941458 + 1.14952i\)
\(L(\frac12)\) \(\approx\) \(0.941458 + 1.14952i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.805 - 1.53i)T \)
7 \( 1 + (-0.969 - 2.46i)T \)
13 \( 1 + (-3.11 + 1.81i)T \)
good5 \( 1 + (-2.64 - 1.52i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.13 - 1.96i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 3.59T + 17T^{2} \)
19 \( 1 + (0.413 + 0.716i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.62iT - 23T^{2} \)
29 \( 1 + (-7.61 + 4.39i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.78 + 8.28i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.34iT - 37T^{2} \)
41 \( 1 + (-7.03 + 4.06i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.70 - 8.14i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.73 - 1.58i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.44 + 0.834i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 10.7iT - 59T^{2} \)
61 \( 1 + (9.14 - 5.28i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.51 - 2.03i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.81 + 6.60i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.74 - 9.94i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.91 + 5.04i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.79iT - 83T^{2} \)
89 \( 1 + 8.34iT - 89T^{2} \)
97 \( 1 + (-7.99 + 13.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.72767117719571425477403517099, −10.11805770296023015373336676162, −9.312788188828209634967318394286, −8.663710234212324070537656159717, −7.76587836236530480428428283684, −6.34861281167380100971082133770, −5.69629397836669131330901505601, −4.43342672388746664122577529231, −2.72866552088610403012451560560, −2.18855942683142421410517064729, 1.10246363137100968479147732291, 1.93124169034097541561281655082, 3.46168411812974896279471108687, 5.13075850666958817846162732168, 6.27644520740412504814803182904, 6.95654679767975215399676114180, 8.084737727484469425537732258784, 8.767450761853996894438356848520, 9.401035592163405991179705944976, 10.53324635601036311703682223653

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