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Properties

Label 2-546-91.81-c1-0-10
Degree $2$
Conductor $546$
Sign $0.899 - 0.437i$
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 + (0.611 + 1.05i)5-s + (−0.5 + 0.866i)6-s + (1.48 − 2.18i)7-s + 0.999·8-s + 9-s − 1.22·10-s + 0.140·11-s + (−0.499 − 0.866i)12-s + (2.39 − 2.69i)13-s + (1.15 + 2.38i)14-s + (0.611 + 1.05i)15-s + (−0.5 + 0.866i)16-s + (−0.0932 − 0.161i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + 0.577·3-s + (−0.249 − 0.433i)4-s + (0.273 + 0.473i)5-s + (−0.204 + 0.353i)6-s + (0.561 − 0.827i)7-s + 0.353·8-s + 0.333·9-s − 0.386·10-s + 0.0423·11-s + (−0.144 − 0.249i)12-s + (0.663 − 0.747i)13-s + (0.307 + 0.636i)14-s + (0.157 + 0.273i)15-s + (−0.125 + 0.216i)16-s + (−0.0226 − 0.0391i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.899 - 0.437i$
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: \((2,\ 546,\ (\ :1/2),\ 0.899 - 0.437i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60343 + 0.369587i\)
\(L(\frac12)\) \(\approx\) \(1.60343 + 0.369587i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 + 2.18i)T \)
13 \( 1 + (-2.39 + 2.69i)T \)
good5 \( 1 + (-0.611 - 1.05i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 0.140T + 11T^{2} \)
17 \( 1 + (0.0932 + 0.161i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 0.895T + 19T^{2} \)
23 \( 1 + (0.0182 - 0.0315i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.99 - 5.18i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.82 - 3.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.181 + 0.314i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.70 - 2.95i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.06 - 3.58i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.358 - 0.621i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.49 + 6.05i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.49 - 6.05i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 0.373T + 61T^{2} \)
67 \( 1 + 4.85T + 67T^{2} \)
71 \( 1 + (5.31 - 9.20i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.80 - 8.32i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.94 + 5.10i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.14T + 83T^{2} \)
89 \( 1 + (-3.17 + 5.50i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.24 + 5.61i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.51039105499043801880529183118, −10.10352848536847853023344083834, −8.866218318292387061879623789056, −8.197773448256686624505221946412, −7.31808510668331049663359568301, −6.56621247291060806373574626546, −5.37891730795254254063911239737, −4.22381193854412866892528132880, −2.97892520154490275290260366635, −1.29087500721472971678750099144, 1.47870498315137075455100032429, 2.51512883128968707311649918591, 3.86264152831529323823771918790, 4.93818141557858093947901995327, 6.09338154799767719982387639169, 7.44057691348614312201401361856, 8.427851648312173454889224642852, 8.985049640242399736337061474582, 9.642890703879383558655428868141, 10.75715816656880886997569022021

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