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Properties

Label 2-546-273.242-c1-0-8
Degree $2$
Conductor $546$
Sign $-0.464 - 0.885i$
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  + (0.965 + 0.258i)2-s + (−0.172 + 1.72i)3-s + (0.866 + 0.499i)4-s + (1.26 + 0.340i)5-s + (−0.612 + 1.62i)6-s + (−1.12 + 2.39i)7-s + (0.707 + 0.707i)8-s + (−2.94 − 0.594i)9-s + (1.13 + 0.657i)10-s + (−0.888 + 0.238i)11-s + (−1.01 + 1.40i)12-s + (−1.58 + 3.23i)13-s + (−1.70 + 2.02i)14-s + (−0.805 + 2.12i)15-s + (0.500 + 0.866i)16-s + (2.11 − 3.66i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.0996 + 0.995i)3-s + (0.433 + 0.249i)4-s + (0.567 + 0.152i)5-s + (−0.250 + 0.661i)6-s + (−0.423 + 0.905i)7-s + (0.249 + 0.249i)8-s + (−0.980 − 0.198i)9-s + (0.360 + 0.207i)10-s + (−0.267 + 0.0717i)11-s + (−0.291 + 0.405i)12-s + (−0.439 + 0.898i)13-s + (−0.455 + 0.541i)14-s + (−0.207 + 0.549i)15-s + (0.125 + 0.216i)16-s + (0.513 − 0.889i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.464 - 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.464 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.464 - 0.885i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.464 - 0.885i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04756 + 1.73231i\)
\(L(\frac12)\) \(\approx\) \(1.04756 + 1.73231i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 + (0.172 - 1.72i)T \)
7 \( 1 + (1.12 - 2.39i)T \)
13 \( 1 + (1.58 - 3.23i)T \)
good5 \( 1 + (-1.26 - 0.340i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.888 - 0.238i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.11 + 3.66i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.896 + 3.34i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.61 - 6.25i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.82iT - 29T^{2} \)
31 \( 1 + (-10.3 + 2.76i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (5.11 + 1.37i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.501 + 0.501i)T - 41iT^{2} \)
43 \( 1 + 2.84iT - 43T^{2} \)
47 \( 1 + (2.66 - 9.93i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.15 - 3.55i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-10.5 + 2.81i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.88 + 6.72i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.69 + 1.52i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-0.0481 + 0.0481i)T - 71iT^{2} \)
73 \( 1 + (0.197 + 0.738i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.43 - 4.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.78 - 5.78i)T - 83iT^{2} \)
89 \( 1 + (-3.09 + 11.5i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (2.51 + 2.51i)T + 97iT^{2} \)
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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

−11.32740862502610015212341658665, −10.02302881742386466490176938938, −9.503705853161273768796862022087, −8.672859186007111715030283031523, −7.27263699013384553058306055963, −6.24484297536237783852112883279, −5.36909194330921278045492153682, −4.71276318299418933346794078031, −3.31188059160779441904958010749, −2.44114904782676552113142109885, 0.983849932793570608747371394782, 2.43546283535136751814873162760, 3.57189807771104190662903975876, 5.02198172994283918493655364714, 5.95997601821681040910934144524, 6.69217748819679882065848228336, 7.68464697501016534670660811236, 8.482356154424890956455390974078, 10.17330019973140175975305672182, 10.31026665295440064798939835587

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