Properties

Label 2-546-273.194-c1-0-9
Degree $2$
Conductor $546$
Sign $0.631 - 0.775i$
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 + (0.278 + 1.70i)3-s + (−0.499 − 0.866i)4-s + (2.44 + 1.41i)5-s + (1.61 + 0.613i)6-s + (−2.60 − 0.449i)7-s − 0.999·8-s + (−2.84 + 0.950i)9-s + (2.44 − 1.41i)10-s + (2.97 + 5.15i)11-s + (1.34 − 1.09i)12-s + (3.07 + 1.87i)13-s + (−1.69 + 2.03i)14-s + (−1.73 + 4.58i)15-s + (−0.5 + 0.866i)16-s + (−2.83 − 4.91i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.160 + 0.987i)3-s + (−0.249 − 0.433i)4-s + (1.09 + 0.632i)5-s + (0.661 + 0.250i)6-s + (−0.985 − 0.169i)7-s − 0.353·8-s + (−0.948 + 0.316i)9-s + (0.774 − 0.447i)10-s + (0.897 + 1.55i)11-s + (0.387 − 0.316i)12-s + (0.854 + 0.520i)13-s + (−0.452 + 0.543i)14-s + (−0.448 + 1.18i)15-s + (−0.125 + 0.216i)16-s + (−0.688 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.631 - 0.775i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.631 - 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70932 + 0.812499i\)
\(L(\frac12)\) \(\approx\) \(1.70932 + 0.812499i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.278 - 1.70i)T \)
7 \( 1 + (2.60 + 0.449i)T \)
13 \( 1 + (-3.07 - 1.87i)T \)
good5 \( 1 + (-2.44 - 1.41i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.97 - 5.15i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.83 + 4.91i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.972 - 1.68i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.43 - 1.40i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.0658iT - 29T^{2} \)
31 \( 1 + (-4.39 - 7.61i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.160 - 0.0928i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.04iT - 41T^{2} \)
43 \( 1 - 3.48T + 43T^{2} \)
47 \( 1 + (6.77 + 3.91i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.77 + 2.18i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-11.6 + 6.71i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (12.8 + 7.39i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.912 + 0.527i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.16T + 71T^{2} \)
73 \( 1 + (0.140 + 0.242i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.18 + 7.24i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.940iT - 83T^{2} \)
89 \( 1 + (4.61 + 2.66i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.60T + 97T^{2} \)
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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

−10.73367806110587508372246040534, −9.962440808163258592589099760043, −9.530640684880265218771625704131, −8.862262935213380219052248987703, −6.90330426453219022715047916757, −6.35298169028943091291013742478, −5.11253740494932133932960993831, −4.12183367247891742148166027910, −3.11214769167932381940650716766, −2.00565501335496533688645094367, 1.03068281958651510851577592870, 2.70142851663622711911766630385, 3.87638172881005430679465254377, 5.64830197538126770048613099096, 6.16265751244326137837132884231, 6.59336427962424840518599306367, 8.136195952902602767163930150793, 8.784289050152357883407358962512, 9.356552566657421709921308936003, 10.76805904390048817534255923755

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