Properties

Label 2-546-91.81-c1-0-8
Degree $2$
Conductor $546$
Sign $0.649 + 0.760i$
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 + (1.10 + 1.91i)5-s + (−0.5 + 0.866i)6-s + (1.44 − 2.21i)7-s − 0.999·8-s + 9-s + 2.20·10-s − 1.05·11-s + (0.499 + 0.866i)12-s + (3.18 + 1.69i)13-s + (−1.19 − 2.36i)14-s + (−1.10 − 1.91i)15-s + (−0.5 + 0.866i)16-s + (0.472 + 0.817i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s − 0.577·3-s + (−0.249 − 0.433i)4-s + (0.493 + 0.854i)5-s + (−0.204 + 0.353i)6-s + (0.547 − 0.836i)7-s − 0.353·8-s + 0.333·9-s + 0.697·10-s − 0.318·11-s + (0.144 + 0.249i)12-s + (0.883 + 0.469i)13-s + (−0.318 − 0.631i)14-s + (−0.284 − 0.493i)15-s + (−0.125 + 0.216i)16-s + (0.114 + 0.198i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.649 + 0.760i$
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: \((2,\ 546,\ (\ :1/2),\ 0.649 + 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51328 - 0.697157i\)
\(L(\frac12)\) \(\approx\) \(1.51328 - 0.697157i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.44 + 2.21i)T \)
13 \( 1 + (-3.18 - 1.69i)T \)
good5 \( 1 + (-1.10 - 1.91i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 1.05T + 11T^{2} \)
17 \( 1 + (-0.472 - 0.817i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 3.92T + 19T^{2} \)
23 \( 1 + (-3.11 + 5.39i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.888 - 1.53i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.63 + 6.29i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.13 - 1.95i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.63 + 2.82i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.537 + 0.930i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.42 - 4.20i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.94 - 8.55i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.509 - 0.882i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 0.0764T + 61T^{2} \)
67 \( 1 + 11.7T + 67T^{2} \)
71 \( 1 + (4.20 - 7.28i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.57 + 11.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.00 - 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.77T + 83T^{2} \)
89 \( 1 + (6.66 - 11.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.99 - 15.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.67664634399130528156796932456, −10.29783012714896928979533206129, −9.182347739887112237132499755446, −7.908815132114264908814499202972, −6.82037267671685146091110351708, −6.07734816525916712785560044804, −4.93514770015376173290559531561, −3.96229893822566388107521417481, −2.68557859451419441553069165072, −1.19594232437629699262268737683, 1.38029416162187881624212788115, 3.21644988148118895484457962904, 4.81119083694822155987677420195, 5.36254202064116434181983872553, 6.01781849271150862305381609895, 7.26142116079835395331683617419, 8.287512902108150806745222464774, 8.987839719526445022529817833606, 9.907393609270671467191730431115, 11.13643618071455743866450259993

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