Properties

Label 2-546-273.101-c1-0-6
Degree $2$
Conductor $546$
Sign $0.00961 - 0.999i$
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 + (−0.320 − 1.70i)3-s + (−0.499 − 0.866i)4-s + (−1.62 + 0.936i)5-s + (1.63 + 0.573i)6-s + (2.47 + 0.928i)7-s + 0.999·8-s + (−2.79 + 1.08i)9-s − 1.87i·10-s − 5.09·11-s + (−1.31 + 1.12i)12-s + (3.20 + 1.64i)13-s + (−2.04 + 1.68i)14-s + (2.11 + 2.46i)15-s + (−0.5 + 0.866i)16-s + (1.48 + 2.56i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.184 − 0.982i)3-s + (−0.249 − 0.433i)4-s + (−0.725 + 0.418i)5-s + (0.667 + 0.234i)6-s + (0.936 + 0.350i)7-s + 0.353·8-s + (−0.931 + 0.363i)9-s − 0.592i·10-s − 1.53·11-s + (−0.379 + 0.325i)12-s + (0.889 + 0.456i)13-s + (−0.545 + 0.449i)14-s + (0.545 + 0.635i)15-s + (−0.125 + 0.216i)16-s + (0.359 + 0.622i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00961 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00961 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.00961 - 0.999i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.00961 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.548166 + 0.542919i\)
\(L(\frac12)\) \(\approx\) \(0.548166 + 0.542919i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.320 + 1.70i)T \)
7 \( 1 + (-2.47 - 0.928i)T \)
13 \( 1 + (-3.20 - 1.64i)T \)
good5 \( 1 + (1.62 - 0.936i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + 5.09T + 11T^{2} \)
17 \( 1 + (-1.48 - 2.56i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 2.60T + 19T^{2} \)
23 \( 1 + (-5.25 - 3.03i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7.30 + 4.21i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.09 - 7.09i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.52 - 3.18i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.85 - 2.80i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.87 - 8.44i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.70 - 1.56i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.21 - 4.74i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.90 + 1.10i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 5.90iT - 61T^{2} \)
67 \( 1 + 11.0iT - 67T^{2} \)
71 \( 1 + (7.27 - 12.6i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.99 - 3.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.14 + 8.91i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.63iT - 83T^{2} \)
89 \( 1 + (12.7 + 7.38i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.83 - 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.12655541262111294624329917106, −10.32764702727196843482254611225, −8.734680592589708566492919073288, −8.149371383411903870681104684249, −7.59440597488786504444135972244, −6.63879813521561063259015853801, −5.66947177705960220481623917448, −4.72443355705407588142041994446, −2.99135128980773373952569346380, −1.47592761839676001830860800779, 0.54798906539594508917783966458, 2.66239987079978777073776832110, 3.87088370144095994060905436332, 4.75709650757058747361876771665, 5.51146025610977459916913931945, 7.29472822086291999853343826500, 8.471822039778547811064486043224, 8.504183678951798293256090364538, 10.01974144945111235750936766854, 10.64131961253473493756656938424

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