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Properties

Label 2-546-273.251-c1-0-32
Degree $2$
Conductor $546$
Sign $-0.923 + 0.382i$
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 + (0.514 − 1.65i)3-s + (−0.499 − 0.866i)4-s − 3.58i·5-s + (1.17 + 1.27i)6-s + (−2.54 + 0.727i)7-s + 0.999·8-s + (−2.47 − 1.70i)9-s + (3.10 + 1.79i)10-s + (−0.630 + 1.09i)11-s + (−1.68 + 0.381i)12-s + (−2.88 − 2.16i)13-s + (0.641 − 2.56i)14-s + (−5.92 − 1.84i)15-s + (−0.5 + 0.866i)16-s + (1.57 + 2.73i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.296 − 0.954i)3-s + (−0.249 − 0.433i)4-s − 1.60i·5-s + (0.479 + 0.519i)6-s + (−0.961 + 0.275i)7-s + 0.353·8-s + (−0.823 − 0.566i)9-s + (0.981 + 0.566i)10-s + (−0.190 + 0.329i)11-s + (−0.487 + 0.110i)12-s + (−0.799 − 0.600i)13-s + (0.171 − 0.686i)14-s + (−1.53 − 0.475i)15-s + (−0.125 + 0.216i)16-s + (0.383 + 0.663i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.923 + 0.382i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.923 + 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.125183 - 0.628963i\)
\(L(\frac12)\) \(\approx\) \(0.125183 - 0.628963i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.514 + 1.65i)T \)
7 \( 1 + (2.54 - 0.727i)T \)
13 \( 1 + (2.88 + 2.16i)T \)
good5 \( 1 + 3.58iT - 5T^{2} \)
11 \( 1 + (0.630 - 1.09i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.57 - 2.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.386 - 0.670i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.28 - 4.20i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.76 + 3.90i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.70T + 31T^{2} \)
37 \( 1 + (0.424 + 0.244i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.17 - 0.676i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.125 - 0.216i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.9iT - 47T^{2} \)
53 \( 1 + 12.1iT - 53T^{2} \)
59 \( 1 + (-10.4 + 6.02i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.27 + 1.88i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.4 + 6.06i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.0759 + 0.131i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.3T + 73T^{2} \)
79 \( 1 - 5.53T + 79T^{2} \)
83 \( 1 + 3.23iT - 83T^{2} \)
89 \( 1 + (5.69 + 3.28i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.21 + 3.83i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.961311158376242988921086323014, −9.265210547436013581104081753994, −8.628793229259664228158124856599, −7.72126385816412108281979367852, −7.01465294979803596978952786727, −5.71465104893146376824897797663, −5.22770550522397746990466571003, −3.55797875701518006263711489104, −1.85432297102164885931674903903, −0.38398769614908862808870759940, 2.66409502730997982053272860050, 3.11111736050616573612202582688, 4.17097410130716652073551044811, 5.57882900459372090002706484388, 6.94088024536049917534622239355, 7.48184327619295632972517422038, 9.066652320347821592721318256729, 9.480168801945506094007489061760, 10.49363403928461064642076238964, 10.80884494122135932369906092947

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