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Properties

Label 2-546-91.81-c1-0-6
Degree $2$
Conductor $546$
Sign $0.841 + 0.540i$
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 + (0.114 + 0.197i)5-s + (−0.5 + 0.866i)6-s + (0.848 + 2.50i)7-s − 0.999·8-s + 9-s + 0.228·10-s + 3.41·11-s + (0.499 + 0.866i)12-s + (−1.62 − 3.21i)13-s + (2.59 + 0.518i)14-s + (−0.114 − 0.197i)15-s + (−0.5 + 0.866i)16-s + (2.70 + 4.68i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s − 0.577·3-s + (−0.249 − 0.433i)4-s + (0.0509 + 0.0883i)5-s + (−0.204 + 0.353i)6-s + (0.320 + 0.947i)7-s − 0.353·8-s + 0.333·9-s + 0.0721·10-s + 1.02·11-s + (0.144 + 0.249i)12-s + (−0.451 − 0.892i)13-s + (0.693 + 0.138i)14-s + (−0.0294 − 0.0509i)15-s + (−0.125 + 0.216i)16-s + (0.656 + 1.13i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.841 + 0.540i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (445, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.841 + 0.540i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49960 - 0.439893i\)
\(L(\frac12)\) \(\approx\) \(1.49960 - 0.439893i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.848 - 2.50i)T \)
13 \( 1 + (1.62 + 3.21i)T \)
good5 \( 1 + (-0.114 - 0.197i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 3.41T + 11T^{2} \)
17 \( 1 + (-2.70 - 4.68i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 6.34T + 19T^{2} \)
23 \( 1 + (-0.959 + 1.66i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.851 - 1.47i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.78 - 3.08i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.09 + 3.62i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.59 - 2.75i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.17 + 8.97i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.57 + 9.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.31 - 5.74i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.38 - 12.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 6.71T + 61T^{2} \)
67 \( 1 - 11.1T + 67T^{2} \)
71 \( 1 + (0.0390 - 0.0677i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.31 - 10.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.811 - 1.40i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.70T + 83T^{2} \)
89 \( 1 + (8.77 - 15.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.82 + 11.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.79480001168168946803668535388, −10.05712847543126562627924751418, −9.132821911319771987559664481773, −8.181261282445747477591025419102, −6.91686206474718405307754818764, −5.78090241961830918285615333942, −5.24609107263644310091375227487, −3.96594555080335639466954088774, −2.75146418175952106417266364340, −1.26052798868033874402035704239, 1.18423878233764156723338362058, 3.35456553261346161861713284635, 4.49251897488558736122646384453, 5.20472399541815381706031323076, 6.40922867739194805293005125874, 7.20346265830511067918135240170, 7.80283760753350300902413073019, 9.397786730907442253612178136107, 9.675055506833174866399249159112, 11.30327054108929564998792543365

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