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Properties

Label 2-546-13.4-c3-0-6
Degree $2$
Conductor $546$
Sign $-0.995 + 0.0894i$
Analytic cond. $32.2150$
Root an. cond. $5.67582$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.73 + i)2-s + (1.5 − 2.59i)3-s + (1.99 + 3.46i)4-s + 19.0i·5-s + (5.19 − 3i)6-s + (−6.06 + 3.5i)7-s + 7.99i·8-s + (−4.5 − 7.79i)9-s + (−19.0 + 33.0i)10-s + (−14.2 − 8.24i)11-s + 12·12-s + (−38.6 + 26.4i)13-s − 14·14-s + (49.5 + 28.6i)15-s + (−8 + 13.8i)16-s + (−5.66 − 9.81i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s + 1.70i·5-s + (0.353 − 0.204i)6-s + (−0.327 + 0.188i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.603 + 1.04i)10-s + (−0.391 − 0.225i)11-s + 0.288·12-s + (−0.825 + 0.564i)13-s − 0.267·14-s + (0.853 + 0.492i)15-s + (−0.125 + 0.216i)16-s + (−0.0808 − 0.140i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.995 + 0.0894i$
Analytic conductor: \(32.2150\)
Root analytic conductor: \(5.67582\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :3/2),\ -0.995 + 0.0894i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.304284981\)
\(L(\frac12)\) \(\approx\) \(1.304284981\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.73 - i)T \)
3 \( 1 + (-1.5 + 2.59i)T \)
7 \( 1 + (6.06 - 3.5i)T \)
13 \( 1 + (38.6 - 26.4i)T \)
good5 \( 1 - 19.0iT - 125T^{2} \)
11 \( 1 + (14.2 + 8.24i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (5.66 + 9.81i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (47.4 - 27.4i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-82.9 + 143. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (120. - 209. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 229. iT - 2.97e4T^{2} \)
37 \( 1 + (-91.0 - 52.5i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (386. + 223. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-166. - 288. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 321. iT - 1.03e5T^{2} \)
53 \( 1 - 362.T + 1.48e5T^{2} \)
59 \( 1 + (550. - 317. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-155. - 269. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (79.5 + 45.9i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (398. - 229. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 736. iT - 3.89e5T^{2} \)
79 \( 1 - 276.T + 4.93e5T^{2} \)
83 \( 1 - 1.27e3iT - 5.71e5T^{2} \)
89 \( 1 + (380. + 219. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (1.21e3 - 700. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−10.91569691702348799785931392594, −10.14551958022590799319905080035, −8.967111150323084446823987353327, −7.83095640214909910328471017563, −6.96462784548470384830904032932, −6.58426740263441026538957681194, −5.49744981872069075185548640871, −4.02988324620382752954574163603, −2.90910193075611938158265238194, −2.28295811498074365339031406461, 0.27807320670283094491732596263, 1.74948461674379315125066924200, 3.16327413254109133568162881807, 4.35726803268824494870806684751, 4.99995288854293403974927283801, 5.78122353395931418294888055637, 7.32779758136724082061982251389, 8.301652323545017105746510800398, 9.264271669207288207163932457534, 9.823787616165774942856797661592

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